Lean Operations and the Toyota Production System 231
the process and saying, “Wow, we have a lot of inventory at this step today”). In contrast,
in a kanban system the amount of inventory becomes a managerial decision variable—the
maximum inventory is controlled via the number of kanban cards in the process.
As an alternative to a kanban system, we also can implement a pull system using
a make-to-order process. As is suggested by the term “make-to-order,” resources in such
a process only operate after having received an explicit customer order. Typically, the
products corresponding to these orders then flow through the process on a first-in, first-out
(FIFO) basis. Each flow unit in the make-to-order process is thereby explicitly assigned to
one specific customer order. Consider the example of a rear-view mirror production in an
auto plant to see the difference between kanban and make-to-order. When the operator in
charge of producing the interior rear-view mirror at the plant receives the work authoriza-
tion through the kanban card, it has not yet been determined which customer order will be
filled with this mirror. All that is known is that there are—at the aggregate—a sufficient
number of customer orders such that production of this mirror is warranted. Most likely,
the final assembly line of the same auto plant (including the mounting of the rear-view
mirror) will be operated in a make-to-order manner, that is, the operator putting in the mir-
ror can see that it will end up in the car of Mr. Smith.
Many organizations use both forms of pull systems. Consider computer maker Dell.
Dell’s computers are configured in work cells. Processes supplying components are often
operated using kanban. Thus, rear-view mirrors at Toyota and power supplies at Dell flow
through the process in sufficient volume to meet customer demand, yet are produced in
response to a kanban card and have not yet been assigned to a specific order.
When considering which form of a pull system one wants to implement, the following
should be kept in mind:
• Kanban should be used for products or parts (a) that are processed in high volume and
limited variety, (b) that are required with a short lead time so that it makes economic
sense to have a limited number of them (as many as we have kanban cards) preproduced,
and (c) for which the costs and efforts related to storing the components are low.
• Make-to-order should be used when (a) products or parts are processed in low volume
and high variety, (b) customers are willing to wait for their order, and (c) it is expensive
or difficult to store the flow units. Chapter 13 will explain the costs and benefits of a
make-to-order production system.
11.5 Quality Management
If we operate with no buffers and want to avoid the waste of rework, operating at zero
defects is a must. To achieve zero defects, TPS relies on defect prevention, rapid defect
detection, and a strong worker responsibility with respect to quality.
Defects can be prevented by “fool-proofing” many assembly operations, that is, by
making mistakes in assembly operations physically impossible (poka-yoke). Components
are designed in a way that there exists one single way of assembling them.
If, despite defect prevention, a problem occurs, TPS attempts to discover and isolate
this problem as quickly as possible. This is achieved through the jidoka concept. The idea
of jidoka is to stop the process immediately whenever a defect is detected and to alert the
line supervisor. This idea goes back to the roots of Toyota as a maker of automated looms.
Just like an automated loom should stop operating in the case of a broken thread, a defec-
tive machine should shut itself off automatically in the presence of a defect.
Shutting down the machine forces a human intervention in the process, which in turn
triggers process improvement (Fujimoto 1999). The jidoka concept has been generalized
to include any mechanism that stops production in response to quality problems, not just
for automated machines. The most well-known form of jidoka is the Andon cord, a cord
232 Chapter 11
running adjacent to assembly lines that enables workers to stop production if they detect
a defect. Just like the jidoka automatic shut-down of machines, this procedure dramatizes
manufacturing problems and acts as a pressure for process improvements.
A worker pulling the Andon cord upon detecting a quality problem is in sharp contrast
to Henry Ford’s historical assembly line that would leave the detection of defects to a final
inspection step. In TPS, “the next step is the customer” and every resource should only let
those flow units move downstream that have been inspected and evaluated as good parts.
Hence, quality inspection is “built in” (tsukurikomi) and happens at every step in the line,
as opposed to relying on a final inspection step alone.
The idea of detect–stop–alert that underlies the jidoka principle is not just a necessity
to make progress towards implementing the zero inventory principle. Jidoka also benefits
from the zero inventory principle, as large amounts of work-in-process inventory achieve
the opposite of jidoka: they delay the detection of a problem, thereby keeping a defective
process running and hiding the defect from the eyes of management. This shows how the
various TPS principles and methods are interrelated, mutually strengthening each other.
To see how work-in-process inventory is at odds with the idea of jidoka, consider a sequence
of two resources in a process, as outlined in Figure 11.4. Assume the activity times at both
resources are equal to one minute per unit. Assume further that the upstream resource (on
the left) suffers quality problems and—at some random point in time—starts producing bad
output. In Figure 11.4, this is illustrated by the resource producing squares instead of circles.
How long will it take until a quality problem is discovered? If there is a large buffer between
the two resources (upper part of Figure 11.4), the downstream resource will continue to
receive good units from the buffer. In this example, it will take seven minutes before the
downstream resource detects the defective flow unit. This gives the upstream resource seven
minutes to continue producing defective parts that need to be either scrapped or reworked.
Thus, the time between when the problem occurred at the upstream resource and the
time it is detected at the downstream resource depends on the size of the buffer between the
two resources. This is a direct consequence of Little’s Law. We refer to the time between
creating a defect and receiving the feedback about the defect as the information turn-
around time (ITAT). Note that we assume in this example that the defect is detected in the
next resource downstream. The impact of inventory on quality is much worse if defects
only get detected at the end of the process (e.g., at a final inspection step). In this case, the
FIGURE 11.4 8 76 1
Information 5432
Turnaround Time
and Its Relationship
with Buffer Size
Defective Unit
Good Unit
4 32 1
Lean Operations and the Toyota Production System 233
ITAT is driven by all inventory downstream from the resource producing the defect. This
motivates the built-in inspection we mentioned above.
11.6 Exposing Problems through Inventory Reduction
Our discussion on quality reveals that inventory covers up problems. So to improve a process,
we need to turn the “inventory hiding quality problems” effect on its head: we want to reduce
inventory to expose defects and then fix the underlying root cause of the defect.
Recall that in a kanban system, the number of kanban cards—and hence the amount of
inventory in the process—is under managerial control. So we can use the kanban system
to gradually reduce inventory and thereby expose quality problems. The kanban system
and its approach to buffers can be illustrated with the following metaphor. Consider a boat
sailing on a canal that has numerous rocks in it. The freight of the boat is very valuable,
so the company operating the canal wants to make sure that the boat never hits a rock.
Figure 11.5 illustrates this metaphor.
One approach to this situation is to increase the water level in the canal. This way, there is
plenty of water over the rocks and the likelihood of an accident is low. In a production setting,
the rocks correspond to quality problems (defects), setup times, blocking or starving, break-
downs, or other problems in the process and the ship hitting a rock corresponds to lost through-
put. The amount of water corresponds to the amount of inventory in the process (i.e., the
number of kanban cards), which brings us back to our previous “buffer-or-suffer” discussion.
An alternative way of approaching the problem is this: instead of covering the rocks with
water, we also could consider reducing the water level in the canal (reduce the number of
kanban cards). This way, the highest rocks are exposed (i.e., we observe a process problem),
which provides us with the opportunity of removing them from the canal. Once this has been
accomplished, the water level is lowered again, until—step by step—all rocks are removed
from the canal. Despite potential short-term losses in throughput, the advantage of this approach
is that it moves the process to a better frontier (i.e., it is better along multiple dimensions).
This approach to inventory reduction is outlined in Figure 11.6. We observe that we first
need to accept a short-term loss in throughput reflecting the reduction of inventory (we stay on
the efficient frontier, as we now have less inventory). Once the inventory level is lowered, we
are able to identify the most prominent problems in the process (rocks in the water). Once iden-
tified, these problems are solved and thereby the process moves to a more desirable frontier.
Both in the metaphor and in our ITAT discussion above, inventory is the key impedi-
ment to learning and process improvement. Since with kanban cards, management is in
FIGURE 11.5 More or Less Inventory? A Simple Metaphor
Source: Stevenson 2006.
Inventory Buffer Argument:
in Process "Increase Inventory"
Toyota Argument:
"Decrease Inventory"
234 Chapter 11 Flow
Rate
FIGURE 11.6
High
Tension between Flow
Rate and Inventory
Levels/ITAT
Increase Inventory Path Advocated
(Smooth Flow) Now by Toyota
Production
Reduce Inventory System
(Blocking or Starving
Becomes More Likely)
Frontier Reflecting New Frontier
Current Process
Low
Low Inventory Inventory
High Inventory (Short ITAT)
(Long ITAT)
control of the inventory level, it can proactively manage the tension between the short-
term need of a high throughput and the long-term objective of improving the process.
11.7 Flexibility
Given that there typically exist fluctuations in demand from the end market, TPS attempts
to create processes with sufficient flexibility to meet such fluctuations. Since forecasts are
more reliable at the aggregate level (across models or components, see discussion of pool-
ing in Chapter 8 and again in Chapter 15), TPS requests workers to be skilled in handling
multiple machines.
• When production volume has to be decreased for a product because of low demand,
TPS attempts to assign some workers to processes creating other products and to have
the remaining workers handle multiple machines simultaneously for the process with
the low-demand product.
• When production volume has to be increased for a product because of high demand, TPS
often uses a second pool of workers (temporary workers) to help out with production.
Unlike the first pool of full-time employees (typically with lifetime employment guarantee
and a broad skill set), these workers are less skilled and can only handle very specific tasks.
Consider the six-step operation shown in Figure 11.7. Assume all activities have an
activity time of one minute per unit. If demand is low (right), we avoid idle time (low aver-
age labor utilization) by running the process with only three operators (typically, full-time
employees). In this case, each operator is in charge of two minutes of work, so we would
achieve a flow rate of 0.5 unit per minute. If demand is high (left in the Figure 11.7), we
assign one worker to each step, that is, we bring in additional (most likely temporary)
workers. Now, the flow rate can be increased to one unit per minute.
This requires that the operators are skilled in multiple assembly tasks. Good training,
job rotation, skill-based payment, and well-documented standard operating procedures
are essential requirements for this. This flexibility also requires that we have a multitiered
workforce consisting of highly skilled full-time employees and a pool of temporary work-
ers (who do not need such a broad skill base) that can be called upon when demand is high.
Such multitask flexibility of workers also can help decrease idle time in cases of activi-
ties that require some worker involvement but are otherwise largely automated. In these
Lean Operations and the Toyota Production System 235
FIGURE 11.7 Multi-task Flexibility
(Note: The figure assumes a 1 minute/unit activity time at each station.)
Takt Time 1 minute Takt Time 2 minutes
Step Step Step Step Step Step Step Step Step Step Step Step
123456 12 34 56
cases, a worker can load one machine and while this machine operates, the worker—
instead of being idle—operates another machine along the process flow (takotei-mochi).
This is facilitated if the process flow is arranged in a U-shaped manner, in which case
a worker can share tasks not only with the upstream and the downstream resource, but also
with another set of tasks in the process. Another important form of flexibility relates to the
ability of one plant to produce more than one vehicle model. Consider the data displayed in
Figure 11.8. The left part of the figure shows how Ford’s vehicles are allocated to Ford’s
production plants. As we can see, many vehicles are dedicated to one plant and many of
the plants can only produce a small set of vehicles. Consequently, if demand increases
relative to the plant’s capacity, that plant is unlikely to have sufficient capacity to fulfill it.
If demand decreases, the plant is likely to have excess capacity.
In an ideal world, the company would be able to make every model in every plant.
This way, high demand from one model would cancel out with low demand from another
one, leading to better plant utilization and more sales. However, such capacity pooling
would require the plants to be perfectly flexible—requiring substantial investments in
production tools and worker skills. An interesting alternative to such perfect flexibility is
the concept of partial flexibility, also referred to as chaining. The idea of chaining is that
every car can be made in two plants and that the vehicle-to-plant assignment creates a
chain that connects as many vehicles and plants as possible. As we will see in Chapter 15,
such partial flexibility results in almost the same benefits of full flexibility, yet at dramati-
cally lower costs. The right side of Figure 11.8 shows the vehicle-to-plant assignment of
FIGURE 11.8 Vehicle-to-Plant Assignments at Ford (Left) and at Nissan (right).
Source: Moreno and Terwiesch (2011).
FUSION MKS MARK LT QUEST
INFINITY OX 56
MILAN Hermosillo SABLE F SERIES Dearborn Truck
MKZ TAURUS Chicago PLATINA TITAN
Aguascallentes
NAVIGATOR Kansas City 2 Canton
ARMADA Smyma
CLIO ALTIMA
Cuernavaca EQUATOR
ESCAPE TAURUS X EXPEDITION Kentucky Truck XTERRA
MARINER Kansas City1 SENTRA MAXIMA
St. Thomas EDGE
CROWN VICTORIA (Ford) VERSA FRONTIER
FLEX
EXPLORER GRAND MARQUIS NISSAN CHASSIS
MOUNTAINEER Louisville MKT
Oakville
TOWN CAR MKX (Ford) NISSAN PICKUP
FOCUS Wayne TSURU
ECONOLINE Avon Lake
PATHFINDER
236 Chapter 11
Nissan (North America) and provides an illustrative example of partial flexibility. In an
environment of volatile demand, this partial flexibility has allowed Nissan to keep its plants
utilized without providing the hefty discounts offered by its competitors.
11.8 Standardization of Work and Reduction of Variability
As we have seen in Chapters 8 and 9, variability is a key inhibitor in our attempt to create
a smooth flow. In the presence of variability, either we need to buffer (which would violate
the zero inventory philosophy) or we suffer occasional losses in throughput (which would
violate the principle of providing the customer with the requested product when demanded).
For this reason, the Toyota Production System explicitly embraces the concepts of variabil-
ity measurement, control, and reduction discussed in Chapter 10.
The need for stability in a JIT process and the vulnerability of an unbuffered process
were visible in the computer industry following the 1999 Taiwanese earthquake. Several
of the Taiwanese factories that were producing key components for computer manufactur-
ers around the world were forced to shut down their production due to the earthquake.
Such an unpredicted shutdown was more disruptive for computer manufacturers with JIT
supply chains than those with substantial buffers (e.g., in the form of warehouses) in their
supply chains (Papadakis 2002).
Besides earthquakes, variability occurs because of quality defects (see above) or because
of differences in activity times for the same or for different operators. Figure 11.9 shows
performance data from a large consumer loan processing organization. The figure compares
the performance of the top-quartile operator (i.e., the operator who has 25 percent of the other
operators achieving a higher performance and 75 percent of the operators achieving a lower
performance) with the bottom quartile operator (the one who has 75 percent of the operators
achieving a higher performance). As we can see, there can exist dramatic differences in the
productivity across employees.
A quartile analysis is a good way to identify the presence of large differences across
operators and to estimate the improvement potential. For example, we could estimate what
would happen to process capacity if all operators would be trained so that they achieve
a performance in line with the current top-quartile performance.
11.9 Human Resource Practices
We have seen seven sources of waste, but the Toyota Production System also refers to an
eighth source—the waste of the human intellect. For this reason, a visitor to an operation
that follows the Toyota Production System philosophy often encounters signs with expres-
sions like “In our company, we all have two jobs: (1) to do our job and (2) to improve it.”
FIGURE 11.9 Underwriting Decisions Percent
(applications per day) Difference
Productivity
Comparison across 15.1 + 66%
Underwriters 9.1
Phone Calls That Reach the 23.5 + 51%
Right Person
(calls per day) 15.6
Top Quartile
Bottom Quartile
Lean Operations and the Toyota Production System 237
To illustrate different philosophies toward workers, consider the following two quotes.
The first one comes from the legendary book Principles of Scientific Management written by
Frederick Taylor, which still makes an interesting read almost a century after its first appear-
ance (once you have read the quote below, you will at least enjoy Taylor’s candid writing
style). The second quote comes from Konosuka Matsushita, the former chairman of Panasonic.
Let us look at Taylor’s opinion first and consider his description of pig iron shoveling,
an activity that Taylor studied extensively in his research. Taylor writes: “This work is so
crude and elementary that the writer firmly believes that it would be possible to train an
intelligent gorilla so as to become a more efficient pig-iron handler than any man can be.”
Now, consider Matsushita, whose quote almost reads like a response to Taylor:
We are going to win and you are going to lose. There is nothing you can do about it, because
the reasons for failure are within yourself. With you, the bosses do the thinking while the
workers wield the screw drivers. You are convinced that this is the way to run a business. For
you, the essence of management is getting the ideas out of the heads of the bosses and in to the
hands of the labour. [. . .] Only by drawing on the combined brainpower of all its employees
can a firm face up to the turbulence and constraints of today’s environment.
TPS, not surprisingly, embraces Matsushita’s perspective of the “combined brainpower.”
We have already seen the importance of training workers as a source of flexibility.
Another important aspect of the human resource practices of Toyota relates to process
improvement. Quality circles bring workers together to jointly solve production problems
and to continuously improve the process (kaizen). Problem solving is very data driven and
follows a standardized process, including control charts, fishbone (Ishikawa) diagrams,
the “Five Whys,” and other problem-solving tools. Thus, not only do we standardize the
production process, we also standardize the process of improvement.
Ishikawa diagrams (also known as fishbone diagrams or cause–effect diagrams)
graphically represent variables that are causally related to a specific outcome, such as an
increase in variation or a shift in the mean. When drawing a fishbone diagram, we typically
start with a horizontal arrow that points at the name of the outcome variable we want to
analyze. Diagonal lines then lead to this arrow representing main causes. Smaller arrows
then lead to these causality lines, creating a fishbonelike shape. An example of this is
given by Figure 11.10. Ishikawa diagrams are simple yet powerful problem-solving tools
that can be used to structure brainstorming sessions and to visualize the causal structure
of a complex system.
A related tool that also helps in developing causal models is known as the “Five Whys.”
The tool is prominently used in Toyota’s organization when workers search for the root
cause of a quality problem. The basic idea of the “Five Whys” is to continually question
(“Why did this happen?”) whether a potential cause is truly the root cause or is merely a
symptom of a deeper problem.
In addition to these operational principles, TPS includes a range of human resource
management practices, including stable employment (“lifetime employment”) for the core
workers combined with the recruitment of temporary workers; a strong emphasis on skill
development, which is rewarded financially through skill-based salaries; and various other
aspects relating to leadership and people management.
11.10 Lean Transformation
How do you turn around an existing operation to achieve operational excellence as we have
discussed it above? Clearly, even an operations management textbook has to acknowledge that
there is more to a successful operational turnaround than the application of a set of tools.
McKinsey, as a consulting firm with a substantial part of its revenues resulting from
operations work, refers to the set of activities required to improve the operations of a client
238 Chapter 11
FIGURE 11.10 Example of an Ishikawa Diagram
Specifications/ Machines
Information
Dimensions Cutting Tool Worn Vise Position Set Incorrectly
Incorrectly
Specified in Clamping Force Machine Tool Coordinates
Drawing Too High or Too Low Set Incorrectly
Dimensions Part Incorrectly Vise Position Shifted
Incorrectly Coded Positioned in Clamp during Production
in Machine Tool
Program Part Clamping Steer Support
Surfaces Corrupted
Height Deviates
Extrusion from Specification
Temperature
Too High
Error in
Measuring Height Extrusion Stock
Undersized
Extrusion
Extrusion Die Rate Too Material
Undersized High Too Soft
People Materials
as a lean transformation. There exist three aspects to such a lean transformation: the operating
system, a management infrastructure, and the mindsets and behaviors of the employees
involved.
With the operating system, the firm refers to various aspects of process management as
we have discussed in this chapter and throughout this book: an emphasis on flow, match-
ing supply with demand, and a close eye on the variability of the process.
But technical solutions alone are not enough. So the operating system needs to be
complemented by a management infrastructure. A central piece of this infrastructure is
performance measurement. Just as we discussed in Chapter 6, defining finance-level per-
formance measures and then cascading them into the operations is a key struggle for many
companies. Moreover, the performance measures should be tracked over time and be made
transparent throughout the organization. The operator needs to understand which perfor-
mance measures he or she is supposed to achieve and how these measures contribute to the
bigger picture. Management infrastructure also includes the development of operator skills
and the establishment of formal problem-solving processes.
Finally, the mindset of those involved in working in the process is central to the
success of a lean transformation. A nurse might get frustrated from operating in an
environment of waste that is keeping him or her from spending time with patients. Yet,
the nurse, in all likelihood, also will be frustrated by the implementation of a new care
process that an outsider imposes on his or her ward. Change management is a topic
well beyond the scope of this book: open communication with everyone involved in
the process, collecting and discussing process data, and using some of the tools dis-
cussed in Chapter 10 as well as with respect to kaizen can help make the transformation
a success.
11.11 Lean Operations and the Toyota Production System 239
Further
Reading Readers who want to learn more about TPS are referred to excellent reading, such as Fujimoto
(1999) or Ohno (1988), from which many of the following definitions are taken.
11.12
Practice Fujimoto (1999) describes the evolution of the Toyota Production System. While not a primary
Problems focus of the book, it also provides excellent descriptions of the main elements of the Toyota Pro-
duction System. The results of the benchmarking studies are reported in Womack, Jones, and Roos
(1991) and Holweg and Pil (2004).
Bohn and Jaikumar (1992) is a classic reading that challenges the traditional, optimization-
focused paradigm of operations management. Their work stipulates that companies should not focus
on optimizing decisions for their existing business processes, but rather should create new processes
that can operate at higher levels of performance.
Drew, McCallum, and Roggenhofer (2004) describe the “Journey to Lean,” a description of the
steps constituting a lean transformation as described by a group of McKinsey consultants.
Tucker (2004) provides a study of TPS-like activities from the perspective of nurses who encoun-
ter quality problems in their daily work. Moreno and Terwiesch discuss flexibility strategies in the
U.S. automotive industry and analyze if and to what extent firms with flexible production systems
are able to achieve higher plant utilization and lower price discounts.
The Wikipedia entries for Toyota, Ford, Industrial Revolution, Gilbreth, and Taylor are also
interesting summaries and were helpful in compiling the historical reviews presented in this
chapter.
Q11.1 (Three Step) Consider a worker-paced line with three process steps, each of which is
staffed with one worker. The sequence of the three steps does not matter for the comple-
tion of the product. Currently, the three steps are operated in the following sequence.
Step Step Step
123
Activity Time 3 4 5
[min./unit]
Q11.2 a. What would happen to the inventory in the process if the process were operated as a
push system?
b. Assuming you would have to operate as a push system, how would you resequence the
three activities?
c. How would you implement a pull system?
(Six Step) Consider the following six-step worker-paced process. Each resource is cur-
rently staffed by one operator. Demand is 20 units per hour. Over the past years, manage-
ment has attempted to rebalance the process, but given that workers can only complete
tasks that are adjacent to each other, no further improvement has been found.
Step Step Step Step Step Step
1 23456
2.8 3 2.7 2.9 1.5
Activity Time 1.4
[min./unit] a. What would you suggest to improve this process? (Hint: Think “out of the box.”)
12Chapter
Betting on Uncertain
Demand: The
Newsvendor Model1
Matching supply and demand is particularly challenging when supply must be chosen
before observing demand and demand is stochastic (uncertain). To illustrate this point,
suppose you are the owner of a simple business: selling newspapers. Each morning you
purchase a stack of papers with the intention of selling them at your newsstand at the cor-
ner of a busy street. Even though you have some idea regarding how many newspapers
you can sell on any given day, you never can predict demand for sure. Some days you sell
all of your papers, while other days end with unsold newspapers to be recycled. As the
newsvendor, you must decide how many papers to buy at the start of each day. Because
you must decide how many newspapers to buy before demand occurs, unless you are very
lucky, you will not be able to match supply to demand. A decision tool is needed to make
the best out of this difficult situation. The newsvendor model is such a tool.
You will be happy to learn that the newsvendor model applies in many more settings
than just the newsstand business. The essential issue is that you must take a firm bet (how
much inventory to order) before some random event occurs (demand) and then you learn
that you either bet too much (demand was less than your order) or you bet too little (demand
exceeded your order). This trade-off between “doing too much” and “doing too little”
occurs in other settings. Consider a technology product with a long lead time to source
components and only a short life before better technology becomes available. Purchase too
many components and you risk having to sell off obsolete technology. Purchase too few
and you may forgo sizable profits. Cisco is a company that can relate to these issues: In
2000 they estimated that they were losing 10 percent of their potential orders to rivals due
to long lead times created by shortages of parts; but by early 2001, the technology bubble
had burst and they had to write off $2.5 billion in inventory.
This chapter begins with a description of the production challenge faced by O’Neill
Inc., a sports apparel manufacturer. O’Neill’s decision also closely resembles the news-
vendor’s task. We then describe the newsvendor model in detail and apply it to O’Neill’s
problem. We also show how to use the newsvendor model to forecast a number of perfor-
mance measures relevant to O’Neill.
1 Data in this chapter have been disguised to protect confidential information.
240
Betting on Uncertain Demand: The Newsvendor Model 241
12.1 O’Neill Inc.
O’Neill Inc. is a designer and manufacturer of apparel, wetsuits, and accessories for
water sports: surf, dive, waterski, wake-board, triathlon, and wind surf. Their product line
ranges from entry-level products for recreational users, to wetsuits for competitive surf-
ers, to sophisticated dry suits for professional cold-water divers (e.g., divers that work on
oil platforms in the North Sea). O’Neill divides the year into two selling seasons: Spring
(February through July) and Fall (August through January). Some products are sold in
both seasons, but the majority of their products sell primarily in a single season. For
example, waterski is active in the Spring season whereas recreational surf products sell
well in the Fall season. Some products are not considered fashionable (i.e., they have little
cosmetic variety and they sell from year to year), for example, standard neoprene black
booties. With product names like “Animal,” “Epic,” “Hammer,” “Inferno,” and “Zen,”
O’Neill clearly also has products that are subject to the whims of fashion. For example,
color patterns on surf suits often change from season to season to adjust to the tastes of the
primary user (15–30-year-old males from California).
O’Neill operates its own manufacturing facility in Mexico, but it does not produce all of its
products there. Some items are produced by the TEC Group, O’Neill’s contract manufacturer
in Asia. While TEC provides many benefits to O’Neill (low cost, sourcing expertise, flexible
capacity, etc.), they do require a three-month lead time on all orders. For example, if O’Neill
orders an item on November 1, then O’Neill can expect to have that item at its distribution
center in San Diego, California, ready for shipment to customers, only on January 31.
To better understand O’Neill’s production challenge, let’s consider a particular wetsuit
used by surfers and newly redesigned for the upcoming spring season, the Hammer 3/2.
(The “3/2” signifies the thickness of the neoprene on the suit: 3 mm thick on the chest and
2 mm everywhere else.) Figure 12.1 displays the Hammer 3/2 and O’Neill’s logo. O’Neill
has decided to let TEC manufacture the Hammer 3/2. Due to TEC’s three-month lead time,
O’Neill needs to submit an order to TEC in November before the start of the spring season.
Using past sales data for similar products and the judgment of its designers and sales rep-
resentatives, O’Neill developed a forecast of 3,200 units for total demand during the spring
season for the Hammer 3/2. Unfortunately, there is considerable uncertainty in that fore-
cast despite the care and attention placed on the formation of the forecast. For example, it
is O’Neill’s experience that 50 percent of the time the actual demand deviates from their
initial forecast by more than 25 percent of the forecast. In other words, only 50 percent of
the time is the actual demand between 75 percent and 125 percent of their forecast.
Although O’Neill’s forecast in November is unreliable, O’Neill will have a much better
forecast for total season demand after observing the first month or two of sales. At that time,
O’Neill can predict whether the Hammer 3/2 is selling slower than forecast, in which case
O’Neill is likely to have excess inventory at the end of the season, or whether the Hammer
3/2 is more popular than predicted, in which case O’Neill is likely to stock out. In the latter
case, O’Neill would love to order more Hammers, but the long lead time from Asia prevents
O’Neill from receiving those additional Hammers in time to be useful. Therefore, O’Neill
essentially must “live or dive” with its single order placed in November.
Fortunately for O’Neill, the economics on the Hammer are pretty good. O’Neill sells
the Hammer to retailers for $190 while it pays TEC $110 per suit. If O’Neill has leftover
inventory at the end of the season, it is O’Neill’s experience that they are able to sell that
inventory for $90 per suit. Figure 12.2 summarizes the time line of events and the econom-
ics for the Hammer 3/2.
So how many units should O’Neill order from TEC? You might argue that O’Neill
should order the forecast for total demand, 3,200, because 3,200 is the most likely outcome.
242 Chapter 12
FIGURE 12.1
O’Neill’s Hammer
3/2 Wetsuit and Logo
for the Surf Market
The forecast is also the value that minimizes the expected absolute difference between the
actual demand and the production quantity; that is, it is likely to be close to the actual demand.
Alternatively, you may be concerned that forecasts are always biased and therefore suggest an
order quantity less than 3,200 would be more prudent. Finally, you might argue that because
FIGURE 12.2 Generate Forecast
of Demand and
Time Line of Events Submit an Order
and Economics for to TEC
O’Neill’s Hammer
3/2 Wetsuit Spring Selling Season
Nov. Dec. Jan. Feb. Mar. Apr. May June July Aug.
Receive Order Leftover Units
from TEC at the Are Discounted
End of the Month
Selling Price = $190
Purchase Price from TEC = $110
Discount Price at End of Season = $90
Betting on Uncertain Demand: The Newsvendor Model 243
the gross margin on the Hammer is more than 40 percent ((190 Ϫ 110)/190 ϭ 0.42), O’Neill
should order more than 3,200 in case the Hammer is a hit. We next define the newsvendor
model and then discuss what the newsvendor model recommends for an order quantity.
12.2 An Introduction to the Newsvendor Model
The newsvendor model considers a setting in which you have only one production or
procurement opportunity. Because that opportunity occurs well in advance of a single sell-
ing season, you receive your entire order just before the selling season starts. Stochastic
demand occurs during the selling season. If demand exceeds your order quantity, then you
sell your entire order. But if demand is less than your order quantity, then you have leftover
inventory at the end of the season.
There is a fixed cost per unit ordered: for the Hammer 3/2, Cost ϭ 110. It is important
that Cost includes only costs that depend on the number of units ordered; amortized fixed
costs should not be included, because they are unaffected by our order quantity decision.
In other words, this cost figure should include all costs that vary with the order quantity
and no costs that do not vary with the order quantity. There is a fixed price for each unit
you sell; in this case, Price ϭ 190.
If there is leftover inventory at the end of the season, then there is some value associ-
ated with that inventory. To be specific, there is a fixed Salvage value that you earn on
each unit of leftover inventory: with the Hammer, the Salvage value ϭ 90. It is possible
that leftover inventory has no salvage value whatsoever, that is, Salvage value ϭ 0. It is
also possible leftover inventory is costly to dispose, in which case the salvage value may
actually be a salvage cost. For example, if the product is a hazardous chemical, then there
is a cost for disposing of leftover inventory; that is, Salvage value < 0 is possible.
To guide your production decision, you need a forecast for demand. O’Neill’s initial
forecast for the Hammer is 3,200 units for the season. But it turns out (for reasons that
are explained later) you need much more than just a number for a forecast. You need to
have a sense of how accurate your forecast is; you need a forecast on your forecast error!
For example, in an ideal world, there would be absolutely no error in your forecast: if the
forecast is 3,200 units, then 3,200 units is surely the demand for the season. In reality, there
will be error in the forecast, but forecast error can vary in size. For example, it is better to
be 90 percent sure demand will be between 3,100 and 3,300 units than it is to be 90 percent
sure demand will be between 2,400 and 4,000 units. Intuition should suggest that you might
want to order a different amount in those two situations.
To summarize, the newsvendor model represents a situation in which a decision maker
must make a single bet (e.g., the order quantity) before some random event occurs (e.g.,
demand). There are costs if the bet turns out to be too high (e.g., leftover inventory that
is salvaged for a loss on each unit). There are costs if the bet turns out to be too low (the
opportunity cost of lost sales). The newsvendor model’s objective is to bet an amount that
correctly balances those opposing forces. To implement the model, we need to identify our
costs and how much demand uncertainty we face. We have already identified our costs, so
the next section focuses on the task of identifying the uncertainty in Hammer 3/2 demand.
12.3 Constructing a Demand Forecast
The newsvendor model balances the cost of ordering too much against the cost of ordering
too little. To do this, we need to understand how much demand uncertainty there is for the
Hammer 3/2, which essentially means we need to be able to answer the following question:
What is the probability demand will be less than or equal to Q units?
244 Chapter 12
for whatever Q value we desire. In short, we need a distribution function. Recall from
statistics, every random variable is defined by its distribution function, F(Q), which is the
probability the outcome of the random variable is Q or lower. In this case the random variable
is demand for the Hammer 3/2 and the distribution function is
F(Q) ϭ Prob{Demand is less than or equal to Q}
For convenience, we refer to the distribution function, F(Q), as our demand forecast
because it gives us a complete picture of the demand uncertainty we face. The objective
of this section is to explain how we can use a combination of intuition and data analysis to
construct our demand forecast.
Distribution functions come in two forms. Discrete distribution functions can be defined
in the form of a table: There is a set of possible outcomes and each possible outcome has a
probability associated with it. The following is an example of a simple discrete distribution
function with three possible outcomes:
Q F(Q)
2,200 0.25
3,200 0.75
4,200 1.00
The Poisson distribution is an example of a discrete distribution function that we will
use extensively. With continuous distribution functions there are an unlimited number of
possible outcomes. Both the exponential and the normal are continuous distribution func-
tions. They are defined with one or two parameters. For example, the normal distribution is
defined by two parameters: its mean and its standard deviation. We use to represent the
mean of the distribution and to represent the standard deviation. ( is the Greek letter mu
and is the Greek letter sigma.) This notation for the mean and the standard deviation is
quite common, so we adopt it here.
In some situations, a discrete distribution function provides the best representation of
demand, whereas in other situations a continuous distribution function works best. Hence,
we work with both types of distribution functions.
Now let’s turn to the complex task of actually creating the forecast. As mentioned in
Section 12.1, the Hammer 3/2 has been redesigned for the upcoming spring season. As a
result, actual sales in the previous season might not be a good guide for expected demand
in the upcoming season. In addition to the product redesign, factors that could influence
expected demand include the pricing and marketing strategy for the upcoming season,
changes in fashion, changes in the economy (e.g., is demand moving toward higher or
lower price points), changes in technology, and overall trends for the sport. To account
for all of these factors, O’Neill surveyed the opinion of a number of individuals in the
organization on their personal demand forecast for the Hammer 3/2. The survey’s results
were averaged to obtain the initial 3,200-unit forecast. This represents the “intuition”
portion of our demand forecast. Now we need to analyze O’Neill’s available data to further
develop the demand forecast.
Table 12.1 presents data from O’Neill’s previous spring season with wetsuits in the
surf category. Notice that the data include both the original forecasts for each product as
well as its actual demand. The original forecast was developed in a process that was com-
parable to the one that led to the 3,200-unit forecast for the Hammer 3/2 for this season.
For example, the forecast for the Hammer 3/2 in the previous season was 1,300 units, but
actual demand was 1,696 units.
Betting on Uncertain Demand: The Newsvendor Model 245
TABLE 12.1 Product Description Forecast Actual Demand Error* A/F Ratio**
Forecasts and Actual
Demand Data for JR ZEN FL 3/2 90 140 Ϫ50 1.56
Surf Wetsuits from EPIC 5/3 W/HD 120 83 37 0.69
the Previous Spring JR ZEN 3/2 140 Ϫ3 1.02
Season WMS ZEN-ZIP 4/3 170 143 7 0.96
HEATWAVE 3/2 170 163 1.25
JR EPIC 3/2 180 212 Ϫ42 0.97
WMS ZEN 3/2 180 175 5 1.08
ZEN-ZIP 5/4/3 W/HOOD 270 195 1.17
WMS EPIC 5/3 W/HD 320 317 Ϫ15 1.15
EVO 3/2 380 369 Ϫ47 1.54
JR EPIC 4/3 380 587 Ϫ49 1.50
WMS EPIC 2MM FULL 390 571 Ϫ207 0.80
HEATWAVE 4/3 430 311 Ϫ191 0.64
ZEN 4/3 430 274 0.56
EVO 4/3 440 239 79 1.42
ZEN FL 3/2 450 623 156 0.81
HEAT 4/3 460 365 191 0.98
ZEN-ZIP 2MM FULL 470 450 Ϫ183 0.25
HEAT 3/2 500 116 1.27
WMS EPIC 3/2 610 635 85 1.36
WMS ELITE 3/2 650 830 10 0.56
ZEN-ZIP 3/2 660 364 354 1.19
ZEN 2MM S/S FULL 680 788 Ϫ135 0.67
EPIC 2MM S/S FULL 740 453 Ϫ220 0.82
EPIC 4/3 1,020 607 286 0.72
WMS EPIC 4/3 1,060 732 Ϫ128 1.46
JR HAMMER 3/2 1,220 1,552 227 0.59
HAMMER 3/2 1,300 721 133 1.30
HAMMER S/S FULL 1,490 1,696 288 1.23
EPIC 3/2 2,190 1,832 Ϫ492 1.60
ZEN 3/2 3,190 3,504 499 0.37
ZEN-ZIP 4/3 3,810 1,195 Ϫ396 0.86
WMS HAMMER 3/2 FULL 6,490 3,289 Ϫ342 0.57
3,673 Ϫ1,314
1,995
521
2,817
*Error ϭ Forecast Ϫ Actual demand
**A/F ratio ϭ Actual demand divided by Forecast
So how does O’Neill know actual demand for a product that stocks out? For example,
how does O’Neill know that actual demand was 1,696 for last year’s Hammer 3/2 if they
only ordered 1,500 units? Because retailers order via phone or fax, O’Neill can keep track
of each retailer’s initial order, that is, the retailer’s demand before the retailer knows a
product is unavailable. (However, life is not perfect: O’Neill’s phone representatives do
not always record a customer’s initial order into the computer system, so there is even
some uncertainty with that figure. We’ll assume this is a minor issue and not address it in
our analysis.) In other settings, a firm may not be able to know actual demand with that
level of precision. For example, a retailer of O’Neill’s products probably does not get
to observe what demand could be for the Hammer 3/2 once the Hammer is out of stock
at the retailer. However, that retailer would know when during the season the Hammer
stocked out and, hence, could use that information to forecast how many additional units
could have been sold during the remainder of the season. Therefore, even if a firm cannot
directly observe lost sales, a firm should be able to obtain a reasonable estimate for what
demand could have been.
246 Chapter 12 Actual
Demand
FIGURE 12.3 7,000
Forecasts and Actual 6,000
Demand for Surf
Wetsuits from the
Previous Season
5,000
4,000
3,000
2,000
1,000
0 1,000 2,000 3,000 4,000 5,000 6,000 7,000
0 Forecast
As can be seen from the data, the forecasts ranged from a low of 90 units to a high of
6,490 units. There was also considerable forecast error: O’Neill goofed with the Women’s
Hammer 3/2 Full suit with a forecast nearly 3,000 units above actual demand, while the
forecast for the Epic 3/2 suit was about 1,300 units too low. Figure 12.3 gives a scatter
plot of forecasts and actual demand. If forecasts were perfect, then all of the observations
would lie along the diagonal line.
While the absolute errors for some of the bigger products are dramatic, the forecast
errors for some of the smaller products are also significant. For example, the actual
demand for the Juniors Zen Flat Lock 3/2 suit was more than 150 percent greater than
forecast. This suggests that we should concentrate on the relative forecast errors instead of
the absolute forecast errors.
Relative forecast errors can be measured with the A/F ratio:
A /F ratio ϭ Actual demand
Forecast
An accurate forecast has an A/F ratio ϭ 1, while an A/F ratio above 1 indicates the fore-
cast was too low and an A/F ratio below 1 indicates the forecast was too high. Table 12.1
displays the A/F ratios for our data in the last column.
Those A/F ratios provide a measure of the forecast accuracy from the previous season.
To illustrate this point, Table 12.2 sorts the data in ascending A/F order. Also included
in the table is each product’s A/F rank in the order and each product’s percentile, the
fraction of products that have that A/F rank or lower. (For example, the product with the
fifth A/F ratio has a percentile of 5/33 ϭ 15.2 percent because it is the fifth product out
of 33 products in the data.) We see from the data that actual demand is less than 80 percent
of the forecast for one-third of the products (the A/F ratio 0.8 has a percentile of 33.3) and
Betting on Uncertain Demand: The Newsvendor Model 247
TABLE 12.2 Product Description Forecast Actual Demand A/F Ratio* Rank Percentile**
Sorted A/F Ratios for
Surf Wetsuits from ZEN-ZIP 2MM FULL 470 116 0.25 1 3.0
the Previous Spring ZEN 3/2 3,190 1,195 0.37 2 6.1
Season ZEN 4/3 0.56 3 9.1
WMS ELITE 3/2 430 239 0.56 4 12.1
WMS HAMMER 3/2 FULL 650 364 0.57 5 15.2
JR HAMMER 3/2 6,490 3,673 0.59 6 18.2
HEATWAVE 4/3 1,220 721 0.64 7 21.2
ZEN 2MM S/S FULL 430 274 0.67 8 24.2
EPIC 5/3 W/HD 680 453 0.69 9 27.3
EPIC 4/3 120 0.72 10 30.3
WMS EPIC 2MM FULL 1,020 83 0.80 11 33.3
ZEN FL 3/2 390 732 0.81 12 36.4
EPIC 2MM S/S FULL 450 311 0.82 13 39.4
ZEN-ZIP 4/3 740 365 0.86 14 42.4
WMS ZEN-ZIP 4/3 3,810 607 0.96 15 45.5
JR EPIC 3/2 170 3,289 0.97 16 48.5
HEAT 4/3 180 163 0.98 17 51.5
JR ZEN 3/2 460 175 1.02 18 54.5
WMS ZEN 3/2 140 450 1.08 19 57.6
WMS EPIC 5/3 W/HD 180 143 1.15 20 60.6
ZEN-ZIP 5/4/3 W/HOOD 320 195 1.17 21 63.6
ZEN-ZIP 3/2 270 369 1.19 22 66.7
HAMMER S/S FULL 660 317 1.23 23 69.7
HEATWAVE 3/2 1,490 788 1.25 24 72.7
HEAT 3/2 170 1,832 1.27 25 75.8
HAMMER 3/2 500 212 1.30 26 78.8
WMS EPIC 3/2 1,300 635 1.36 27 81.8
EVO 4/3 610 1,696 1.42 28 84.8
WMS EPIC 4/3 440 830 1.46 29 87.9
JR EPIC 4/3 1,060 623 1.50 30 90.9
EVO 3/2 380 1,552 1.54 31 93.9
JR ZEN FL 3/2 380 571 1.56 32 97.0
EPIC 3/2 587 1.60 33 100.0
90 140
2,190 3,504
*A/F ratio ϭ Actual demand divided by Forecast
**Percentile ϭ Rank divided by total number of wetsuits (33)
actual demand is greater than 125 percent of the forecast for 27.3 percent of the products
(the A/F ratio 1.25 has a percentile of 72.7).
Given that the A/F ratios from the previous season reflect forecast accuracy in the
previous season, maybe the current season’s forecast accuracy will be comparable.
Hence, we want to find a distribution function that will match the accuracy we observe
in Table 12.2. We will use the normal distribution function to do this. Before getting
there, we need a couple of additional results.
Take the definition of the A/F ratio and rearrange terms to get
Actual demand A /F ratio Forecast
For the Hammer 3/2, the forecast is 3,200 units. Note that the forecast is not random,
but the A/F ratio is random. Hence, the randomness in actual demand is directly related
248 Chapter 12
to the randomness in the A/F ratio. Using standard results from statistics and the above
equation, we get the following results:
Expected actual demand Expected A/F ratio Forecast
and
Standard deviation of demand Standard deviattion of A/F ratios Forecast
Expected actual demand, or expected demand for short, is what we should choose
for the mean for our normal distribution, . The average A/F ratio in Table 12.2 is
0.9976. Therefore, expected demand for the Hammer 3/2 in the upcoming season is
0.9976 ϫ 3,200 ϭ 3,192 units. In other words, if the initial forecast is 3,200 units and the
future A/F ratios are comparable to the past A/F ratios, then the mean of actual demand is
3,192 units. So let’s choose 3,192 units as our mean of the normal distribution.
This decision may raise some eyebrows: If our initial forecast is 3,200 units, why do
we not instead choose 3,200 as the mean of the normal distribution? Because 3,192 is so
close to 3,200, assigning 3,200 as the mean probably would lead to a good order quantity
as well. However, suppose the average A/F ratio were 0.90, that is, on average, actual
demand is 90 percent of the forecast. It is quite common for people to have overly opti-
mistic forecasts, so an average A/F ratio of 0.90 is possible. In that case, expected actual
demand would only be 0.90 ϫ 3,200 ϭ 2,880. Because we want to choose a normal distri-
bution that represents actual demand, in that situation it would be better to choose a mean
of 2,880 even though our initial forecast is 3,200. (Novice golfers sometimes adopt an
analogous strategy. If a golfer consistently hooks the ball to the right on her drives, then
she should aim to the left of the flag. In an ideal world, there would be no hook to her shot
nor a bias in the forecast. But if the data say there is a hook, then it should not be ignored.
Of course, the golfer and the forecaster also should work on eliminating the bias.)
Now that we have a mean for our normal distribution, we need a standard deviation. The sec-
ond equation above tells us that the standard deviation of actual demand equals the standard devi-
ation of the A/F ratios times the forecast. The standard deviation of the A/F ratios in Table 12.2
is 0.369. (Use the “stdev()” function in Excel.) So the standard deviation of actual demand
is the standard deviation of the A/F ratios times the initial forecast: 0.369 ϫ 3,200 ϭ 1,181.
Hence, to express our demand forecast for the Hammer 3/2, we can use a normal distribution
with a mean of 3,192 and a standard deviation of 1,181. See Exhibit 12.1 for a summary of the
process of choosing a mean and a standard deviation for a normal distribution forecast.
Now that we have the parameters of a normal distribution that will express our demand
forecast, we need to be able to find F(Q). There are two ways this can be done. The first
way is to use spreadsheet software. For example, in Excel use the function Normdist(Q,
3192, 1181, 1). The second way, which does not require a computer, is to use the Standard
Normal Distribution Function Table in Appendix B.
The standard normal is a particular normal distribution: its mean is 0 and its standard
deviation is 1. To introduce another piece of common Greek notation, let ⌽(z) be the dis-
tribution function of the standard normal. Even though the standard normal is a continuous
distribution, it can be “chopped up” into pieces to make it into a discrete distribution. The
Standard Normal Distribution Function Table is exactly that; that is, it is the discrete ver-
sion of the standard normal distribution. The full table is in Appendix B, but Table 12.3
reproduces a portion of the table.
The format of the Standard Normal Distribution Function Table makes it somewhat tricky
to read. For example, suppose you wanted to know the probability that the outcome of a stan-
dard normal is 0.51 or lower. We are looking for the value of ⌽(z) with z ϭ 0.51. To find that
value, pick the row and column in the table such that the first number in the row and the first
number in the column add up to the z value you seek. With z ϭ 0.51, we are looking for the
row that begins with 0.50 and the column that begins with 0.01, because the sum of those two
Exhibit 12.1
A PROCESS FOR USING HISTORICAL A/F RATIOS TO CHOOSE A MEAN
AND STANDARD DEVIATION FOR A NORMAL DISTRIBUTION FORECAST
Step 1. Assemble a data set of products for which the forecasting task is comparable to the
product of interest. In other words, the data set should include products that you
expect would have similar forecast error to the product of interest. (They may or
may not be similar products.) The data should include an initial forecast of demand
and the actual demand. We also need forecast for the item for the upcoming season.
Step 2. Evaluate the A/F ratio for each product in the data set. Evaluate the average of the
A/F ratios (that is, the expected A/F ratio) and the standard deviation of the A/F ratios.
(In Excel use the average() and stdev() functions.)
Step 3. The mean and standard deviation of the normal distribution that we will use as
the forecast can now be evaluated with the following two equations:
Expected demand Expected A/F ratio Forecast
Standard deviation of demand Standard deviation of A/F ratios Forecast
where the forecast in the above equations is the forecast for the item for the
upcoming season.
values equals 0.51. The intersection of that row with that column gives ⌽(z); from Table 12.3
we see that ⌽(0.51) ϭ 0.6950. Therefore, there is a 69.5 percent probability the outcome of a
standard normal is 0.51 or lower.
But it is unlikely that our demand forecast will be a standard normal distribution. So how
can we use the standard normal to find F(Q); that is, the probability demand will be Q or
lower given that our demand forecast is some other normal distribution? The answer is that
we convert the quantity we are interested in, Q, into an equivalent quantity for the standard
normal. In other words, we find a z such that F(Q) ϭ ⌽(z); that is, the probability demand
is less than or equal to Q is the same as the probability the outcome of a standard normal is
z or lower. That z is called the z-statistic. Once we have the appropriate z-statistics, we then
just look up ⌽(z) in the Standard Normal Distribution Function Table to get our answer.
To convert Q into the equivalent z-statistic, use the following equation:
z ϭ Q Ϫ m
s
For example, suppose we are interested in the probability that demand for the Hammer 3/2
will be 4,000 units or lower, that is, Q ϭ 4,000. With a normal distribution that has mean
3,192 and standard deviation 1,181, the quantity Q ϭ 4,000 has a z-statistic of
z 4, 000 3,192 0.68
1,181
TABLE 12.3 z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
A Portion of the
Standard Normal 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
Distribution Function 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
Table, ⌽(z) 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8269 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
249
Exhibit 12.2
A PROCESS FOR EVALUATING THE PROBABILITY DEMAND IS EITHER LESS THAN
OR EQUAL TO Q (WHICH IS F(Q )) OR MORE THAN Q ( WHICH IS 1 ؊ F(Q))
If the demand forecast is a normal distribution with mean and standard deviation , then
follow steps A and B:
A. Evaluate the z-statistic that corresponds to Q:
z ϭ Q Ϫ m
s
B. The probability demand is less than or equal to Q is ⌽(z). With Excel ⌽(z) can be evalu-
ated with the function Normsdist(z); otherwise, look up ⌽(z) in the Standard Normal
Distribution Function Table in Appendix B. If you want the probability demand is greater
than Q, then your answer is 1 Ϫ ⌽(z).
If the demand forecast is a discrete distribution function table, then look up F(Q), which is
the probability demand is less than or equal to Q. If you want the probability demand is
greater than Q, then the answer is 1 Ϫ F(Q).
Therefore, the probability demand for the Hammer 3/2 is 4,000 units or lower is ⌽(0.68);
that is, it is the same as the probability the outcome of a standard normal is 0.68 or lower.
According to the Standard Normal Distribution Function Table (see Table 12.3 for conve-
nience), ⌽(0.68) ϭ 0.7517. In other words, there is just over a 75 percent probability that
demand for the Hammer 3/2 will be 4,000 or fewer units. Exhibit 12.2 summarizes the pro-
cess of finding the probability demand will be less than or equal to some Q (or more than Q).
You may recall that it has been O’Neill’s experience that demand deviated by more than
25 percent from their initial forecast for 50 percent of their products. We can now check
whether that experience is consistent with our normal distribution forecast for the Hammer
3/2. Our initial forecast is 3,200 units. So a deviation of 25 percent or more implies demand
is either less than 2,400 units or more than 4,000 units. The z-statistic for Q ϭ 2,400 is
z ϭ (2400 − 3192)/1181 ϭ −0.67, and from the Standard Normal Distribution Function
Table, ⌽(−0.67) ϭ 0.2514. (Find the row with Ϫ0.60 and the column with −0.07.) If there
is a 25.14 percent probability demand is less than 2,400 units and a 75.17 percent prob-
ability that demand is less than 4,000 units, then there is a 75.17 − 25.14 ϭ 50.03 percent
probability that demand is between 2,400 and 4,000 units. Hence, O’Neill’s initial assertion
regarding forecast accuracy is consistent with our normal distribution forecast of demand.
To summarize, the objective in this section is to develop a detailed demand forecast. A
single “point forecast” (e.g., 3,200 units) is not sufficient. We need to quantify the amount
of variability that may occur about our forecast; that is, we need a distribution function.
We obtained this distribution function by fitting a normal distribution to our historical
forecast accuracy data, Table 12.2.
12.4 The Expected Profit-Maximizing Order Quantity
The next step after assembling all of our inputs (selling price, cost, salvage value, and
demand forecast) is to choose an order quantity. The first part in that process is to decide
what is our objective. A natural objective is to choose our production/procurement quantity
250
Betting on Uncertain Demand: The Newsvendor Model 251
to maximize our expected profit. This section explains how to do this. Section 12.6 consid-
ers other possible objectives.
Before revealing the actual procedure for choosing an order quantity to maximize
expected profit, it is helpful to explore the intuition behind the solution. Consider again
O’Neill’s Hammer 3/2 ordering decision. Should we order one unit? If we do, then there is
a very good chance we will sell the unit: With a forecast of 3,192 units, it is likely we sell
at least one unit. If we sell the unit, then the gain from that unit equals $190 Ϫ $110 ϭ $80
(the selling price minus the purchase cost). The expected gain from the first unit, which
equals the probability of selling the first unit times the gain from the first unit, is then very
close to $80. However, there is also a slight chance that we do not sell the first unit, in
which case we incur a loss of $110 Ϫ $90 ϭ $20. (The loss equals the difference between
the purchase cost and the discount price.) But since the probability we do not sell that unit
is quite small, the expected loss on the first unit is nearly $0. Given that the expected gain
from the first unit clearly exceeds the expected loss, the profit from ordering that unit is
positive. In this case it is a good bet to order at least one unit.
After deciding whether to order one unit, we can now consider whether we should
order two units, and then three units, and so forth. Two things happen as we continue this
process. First, the probability that we sell the unit we are considering decreases, thereby
reducing the expected gain from that unit. Second, the probability we do not sell that unit
increases, thereby increasing the expected loss from that unit. Now imagine we order the
6,400th unit. The probability of selling that unit is quite low, so the expected gain from that
unit is nearly zero. In contrast, the probability of not selling that unit is quite high, so the
expected loss is nearly $20 on that unit. Clearly it makes no sense to order the 6,400th unit.
This pattern is illustrated in Figure 12.4. We see that from some unit just above 4,000 the
expected gain on that unit equals its expected loss.
Let’s formalize this intuition some more. In the newsvendor model, there is a trade-off
between ordering too much (which could lead to costly leftover inventory) and ordering
too little (which could lead to the opportunity cost of lost sales). To balance these forces,
it is useful to think in terms of a cost for ordering too much and a cost for ordering too
little. Maximizing expected profit is equivalent to minimizing those costs. To be specific,
let Co be the overage cost, the loss incurred when a unit is ordered but not sold. In other
words, the overage cost is the per-unit cost of overordering. For the Hammer 3/2, we
have Co ϭ 20.
FIGURE 12.4 Expected gain or loss 90 Expected gain
80
The Expected Gain 70 Expected loss
and Expected Loss 60
from the Qth 50 800 1600 2400 3200 4000 4800 5600 6400
Hammer 3/2 Ordered 40 Qth unit ordered
by O’Neill 30
20
10
0
0
252 Chapter 12
In contrast to Co, let Cu be the underage cost, the opportunity cost of not ordering a unit
that could have been sold. The following is an equivalent definition for Cu: Cu is the gain
from selling a unit. In other words, the underage cost is the per-unit opportunity cost of
underordering. For the Hammer 3/2, Cu ϭ 80. Note that the overage and underage costs are
defined for a single unit. In other words, Co is not the total cost of all leftover inventory;
instead, Co is the cost per unit of leftover inventory. The reason for defining Co and Cu for a
single unit is simple: We don’t know how many units will be left over in inventory, or how
many units of demand will be lost, but we do know the cost of each unit left in inventory
and the opportunity cost of each lost sale.
Now that we have defined the overage and underage costs, we need to choose Q to
strike the balance between them that results in the maximum expected profit. Based on our
previous reasoning, we should keep ordering additional units until the expected loss equals
the expected gain.
The expected loss on a unit is the cost of having the unit in inventory (the overage cost)
times the probability it is left in inventory. For the Qth unit, that probability is F(Q): It is
left in inventory if demand is less than Q.2 Therefore, the expected loss is Co ϫ F(Q). The
expected gain on a unit is the benefit of selling a unit (the underage cost) times the proba-
bility the unit is sold, which in this case occurs if demand is greater than Q. The probability
demand is greater than Q is (1 Ϫ F(Q)). Therefore, the expected gain is Cu ϫ (1 Ϫ F(Q)).
It remains to find the order quantity Q that sets the expected loss on the Qth unit equal
to the expected gain on the Qth unit:
Co F(Q) Cu (1 F(Q))
If we rearrange terms in the above equation, we get
F (Q) Cu (12.1)
Co Cu
The profit-maximizing order quantity is the order quantity that satisfies the above equa-
tion. If you are familiar with calculus and would like to see a more mathematically rigor-
ous derivation of the optimal order quantity, see Appendix D.
So how can we use equation (12.1) to actually find Q? Let’s begin by just reading it.
It says that the order quantity that maximizes expected profit is the order quantity Q such
that demand is less than or equal to Q with probability Cu/(Co ϩ Cu). That ratio with the
underage and overage costs is called the critical ratio. We now have an explanation for
why our forecast must be a distribution function. To choose the profit-maximizing order
quantity, we need to find the quantity such that demand will be less than that quantity with
a particular probability (the critical ratio). The mean alone (i.e., just a sales forecast) is
insufficient to do that task.
Let’s begin with the easy part. We know for the Hammer 3/2 that Cu ϭ 80 and Co ϭ 20,
so the critical ratio is
Cu 80 0.8
Co Cu 20 80
2 That statement might bother you. You might recall that F(Q) is the probability demand is Q or
lower. If demand is exactly Q, then the Qth unit will not be left in inventory. Hence, you might argue
that it is more precise to say that F(Q Ϫ 1) is the probability the Qth unit is left in inventory. However,
the normal distribution assumes demand can be any value, including values that are not integers. If
you are willing to divide each demand into essentially an infinite number of fractional pieces, as is
assumed by the normal, then F(Q) is indeed the probability there is leftover inventory. If you are
curious about the details, see Appendix D.
Betting on Uncertain Demand: The Newsvendor Model 253
We are making progress, but now comes the tricky part: We need to find the order quantity
Q such that there is a 80 percent probability that demand is Q or lower.
There are two ways to find a Q such that there is an 80 percent probability that demand
will be Q or smaller. The first is to use the Excel function, Normsinv(), and the second is to
use the Standard Normal Distribution Function Table. If you have Excel available, the first
method is the easiest, but they both follow essentially the same process, as we will see.
If we have Excel, to find the optimal Q, we begin by finding the z statistic such that
there is an 80 percent probability the outcome of a standard normal is z or lower. Then we
convert that z into the Q we seek. To find our desired z, use the following Excel function:
z ϭ Normsinv(Critical ratio)
In our case, the critical ratio is 0.80 and Normsinv(0.80) returns 0.84. That means that
there is an 80 percent chance the outcome of a standard normal will be 0.84 or lower. That
would be our optimal order quantity if demand followed a standard normal distribution.
But our demand is not standard normal. It is normal with mean 3192 and standard devia-
tion 1181. To convert our z into an order quantity that makes sense for our actual demand
forecast, we use the following equation:
where Q ϭmϩzϫs
m ϭ Mean of the normal distribution
s ϭ Standard deviation of the normal distribution
Hence, using our Excel method, the expected profit maximizing order quantity for the
Hammer 3/2 is Q ϭ 3,192 ϩ 0.84 ϫ 1,181 ϭ 4,184.
The second method to find Q is to use the Standard Normal Distribution Function
Table. Again, we want to find the z such that the probability the standard normal is z or
less is equal to the critical ratio, which in this case is 0.80. Looking at Table 12.3, we see
that ⌽(0.84) ϭ 0.7995 and ⌽(0.85) ϭ 0.8023, neither of which is exactly the 0.80 prob-
ability we are looking for: z ϭ 0.84 yields a slightly lower probability (79.95 percent) and
z ϭ 0.85 yields a slightly higher probability (80.23 percent). What should we do? The rule
is simple, which we will call the round-up rule:
Round-up rule. Whenever you are looking up a target value in a table and the target
value falls between two entries, choose the entry that leads to the
larger order quantity.
In this case the larger quantity is z ϭ 0.85, so we will go with 0.85. Now, like with our
Excel process, we convert that z into a Q ϭ 3,192 ϩ 0.85 ϫ 1,181 ϭ 4,196.
Why do our two methods lead to different answers? In short, Excel does not implement
the round-up rule. But that raises the next question. Is it OK to use Excel to get our answer?
The answer is “yes.” To explain, when demand is normally distributed, there will be a small
difference between the Excel answer, using the Normsinv() function, and the Standard Nor-
mal Distribution Function Table answer. In this case, the difference between the two is only
12 units, which is less than 0.3 percent away from 4,196.
Therefore, the expected profit with either of these order quantities will be essentially
the same. Furthermore, Excel provides a convenient means to perform this calculation
quickly.
So, if Excel is the quick and easy method, why should we bother with the Standard
Normal Distribution Function Table and the round-up rule? Because when our demand
forecast is a discrete distribution function, the round-up rule provides the more accurate
answer. (Recall, a discrete distribution function assumes that the only possible outcomes
Exhibit 12.3
A PROCEDURE TO FIND THE ORDER QUANTITY THAT MAXIMIZES EXPECTED PROFIT
IN THE NEWSVENDOR MODEL
Step 1: Evaluate the critical ratio: Cu . In the case of the Hammer 3/2, the underage
Co ϩ Cu
cost is Cu ϭ Price Ϫ Cost and the overage cost is Co ϭ Cost Ϫ Salvage value.
Step 2: If the demand forecast is a normal distribution with mean and standard deviation
, then follow steps A and B:
A. Find the optimal order quantity if demand had a standard normal distribu-
tion. One method to achieve this is to find the z value in the Standard Normal
Distribution Function Table such that
F(z) Cu
Co Cu
(If the critical ratio value does not exist in the table, then find the two z values that
it falls between. For example, the critical ratio 0.80 falls between z ϭ 0.84 and
z ϭ 0.85. Then choose the larger of those two z values.) A second method is to
use the Excel function Normsinv: z ϭ Normsinv(Critical ratio).
B. Convert z into the order quantity that maximizes expected profit, Q:
Qϭϩzϫ
are integers.) This is particularly valuable when expected demand is small, say, 10 units,
or 1 unit, or even 0.25 unit. In those cases, the normal distribution function does not model
demand well (in part, because it is a continuous distribution function). Furthermore, it can
make a big difference (in terms of expected profit) whether one or two units are ordered.
Hence, the value of understanding the round-up rule.
This discussion probably has left you with one final question—Why is the round-up
rule the right rule? The critical ratio is actually closer to 0.7995 (which corresponds to
z ϭ 0.84) than it is to 0.8023 (which corresponds to z ϭ 0.85). That is why Excel chooses
z ϭ 0.84. Shouldn’t we choose the z value that leads to the probability that is closest to
the critical ratio? In fact, that is not the best approach. The critical ratio equation works
with the following logic—keep ordering until you get to the first order quantity such that
the critical ratio is less than the probability demand is that order quantity or lower. That
logic leads the rule to “step over” the critical ratio and then stop, that is, the round-up rule.
Excel, in contrast, use the “get as close to the critical ratio as possible” rule. If you are
hungry for a more in-depth explanation and justification, see Appendix D. Otherwise, stick
with the round-up rule, and you will be fine. Exhibit 12.3 summarizes these steps.
12.5 Performance Measures
The previous section showed us how to find the order quantity that maximizes our expected
profit. This section shows us how to evaluate a number of relevant performance measures.
As Figure 12.5 indicates, these performance measures are closely related. For example,
to evaluate expected leftover inventory, you first evaluate expected lost sales (which has
up to three inputs: the order quantity, the loss function table, and the standard deviation
of demand), then expected sales (which has two inputs: expected lost sales and expected
demand), and then expected leftover inventory (which has two inputs: expected sales and
the order quantity).
254
Betting on Uncertain Demand: The Newsvendor Model 255
FIGURE 12.5 Expected Expected Expected Expected
Demand, m Lost Sales Sales Profit
The Relationships
between Initial Input If Normal In-Stock Expected
Parameters (boxes) Demand, s Probability Leftover
and Performance Inventory
Measures (ovals) Loss Function
Note: Some Table
performance measures
require other Order Quantity,
performance measures Q, and, if
as inputs; for example,
expected sales Normal Demand,
requires expected z = (Q – m)/s
demand and expected
lost sales as inputs. Distribution
Function Table
Stockout
Price, Cost, Probability
Salvage Value
These performance measures can be evaluated for any order quantity, not just the
expected profit-maximizing order quantity. To emphasize this point, this section evaluates
these performance measures assuming 3,500 Hammer 3/2s are ordered. See Table 13.1 for
the evaluation of these measures with the optimal order quantity, 4,196 units.
Expected Lost Sales
Let’s begin with expected lost sales, which is the expected number of units by which
demand (a random variable) exceeds the order quantity (a fixed threshold). (Because order
quantities are measured in physical units, sales and lost sales are measured in physical
units as well, not in monetary units.) For example, if we order 3,500 units of the Ham-
mer but demand could have been high enough to sell 3,821 units, then we would lose
3,821 − 3,500 ϭ 321 units of demand. Expected lost sales is the amount of demand that is
not satisfied, which should be of interest to a manager even though the opportunity cost of
lost sales does not show up explicitly on any standard accounting document.
Note that we are interested in the expected lost sales. Demand can be less than our
order quantity, in which case lost sales is zero, or demand can exceed our order quantity,
in which case lost sales is positive. Expected lost sales is the average of all of those events
(the cases with no lost sales and all cases with positive lost sales).
How do we find expected lost sales for any given order quantity? When demand is
normally distributed, use the following equation:
where Expected lost sales ϭ s ϫ L(z)
s ϭ Standard deviation of the normal distribution representing demand
L(z) ϭ Loss function with the standard normal distribution
256 Chapter 12
We already know ϭ 1,181 but what is L(z)? Like with the optimal order quantity, there
are two methods to find L(z), one using Excel and one using a table. With either method, we
first find the z-statistic that corresponds to our chosen order quantity, Q ϭ 3,500:
z ϭ Q Ϫ m ϭ 3, 500 Ϫ 3,192 ϭ 0.26
s 1,181
The first method then uses the following Excel formula to evaluate the expected lost
sales if demand were a standard normal distribution, L(z):
L(z) Normdist(z,0,1,0) z*(1 Normsdist (z))
(If you are curious about the derivation of the above function, see Appendix D.) In this case,
Excel provides the following answer: L(0.26) ϭ Normdist(0.26,0,1,0) Ϫ 0.26 *(1 Ϫ
Normsdist(0.26)) = 0.2824.
The second method uses the Standard Normal Loss Function Table in Appendix B to
look up the expected lost sales. From that table we see that L(0.26) ϭ 0.2824. In this case,
our two methods yield the same value for L(z), which always is the case when we input
into the Excel function a z value rounded to the nearest hundredth (e.g., 0.26 instead of
0.261). Therefore, if the order quantity is 3,500 Hammer 3/2s, then we can expect to lose
ϫ L(z) ϭ 1,181 ϫ 0.2824 = 334 units of demand.
How do we evaluate expected lost sales when we do not use a normal distribution to
model demand? In that situation we need a table to tell us what expected lost sales is for
our chosen order quantity. For example, Appendix B provides the loss function for the
Poisson distribution with different means. Appendix C provides a procedure to evaluate
the loss function for any discrete distribution function. We relegate this procedure to the
appendix because it is computationally burdensome; that is, it is the kind of calculation
you want to do on a spreadsheet rather than by hand.
Exhibit 12.4 summarizes the procedures for evaluating expected lost sales.
Expected Sales
Each unit of demand results in either a sale or a lost sale, so
Expected sales Expected lost sales Expected demand
We already know expected demand: It is the mean of the demand distribution, . Rear-
range terms in the above equation and we get
Expected sales ϭ m Ϫ Expected lost sales
Therefore, the procedure to evaluate expected sales begins by evaluating expected lost
sales. See Exhibit 12.5 for a summary of this procedure.
Let’s evaluate expected sales if 3,500 Hammers are ordered and the normal distribution
is our demand forecast. We already evaluated expected lost sales to be 334 units. There-
fore, Expected sales ϭ 3,192 − 334 ϭ 2,858 units.
Notice that expected sales is always less than expected demand (because expected lost
sales is never negative). While you might get lucky and sell more than the mean demand,
on average you cannot sell more than the mean demand.
Exhibit 12.4
EXPECTED LOST SALES EVALUATION PROCEDURE
If the demand forecast is a normal distribution with mean and standard deviation , then
follow steps A through D:
A. Evaluate the z-statistic for the order quantity Q: z ϭ QϪm .
s
B. Use the z-statistic to look up in the Standard Normal Loss Function Table the expected
lost sales, L(z), with the standard normal distribution.
C. Expected lost sales ϭ ϫ L(z).
D. With Excel, expected lost sales can be evaluated with the following equation:
Expected lost sales ϭ s*(Normdist(z,0,1,0) Ϫ z*(1 Ϫ Normsdist(z)))
If the demand forecast is a discrete distribution function table, then expected lost sales
equals the loss function for the chosen order quantity, L(Q). If the table does not include
the loss function, then see Appendix C for how to evaluate it.
Expected Leftover Inventory
Expected leftover inventory is the average amount that demand (a random variable) is less
than the order quantity (a fixed threshold). (In contrast, expected lost sales is the average
amount by which demand exceeds the order quantity.)
The following equation is true because every unit purchased is either sold or left over in
inventory at the end of the season:
Expected sales Expected leftover inventory Q
Rearrange the above equation to obtain
Expected leftover inventory Q Expected sales
We know the quantity purchased, Q. Therefore, we can easily evaluate expected leftover inven-
tory once we have evaluated expected sales. See Exhibit 12.5 for a summary of this procedure.
If the demand forecast is a normal distribution and 3,500 Hammers are ordered, then
expected leftover inventory is 3,500 − 2,858 ϭ 642 units because we evaluated expected
sales to be 2,858 units.
It may seem surprising that expected leftover inventory and expected lost sales can both
be positive. While in any particular season there is either leftover inventory or lost sales,
but not both, we are interested in the expectation of those measures over all possible out-
comes. Therefore, each expectation can be positive.
Expected Profit
We earn Price − Cost on each unit sold and we lose Cost − Salvage value on each unit we
do not sell, so our expected profit is
Expected profit [(Price Cost) Expected sales]
[(Cost Salvage value) Expected leftover inventory]
Therefore, we can evaluate expected profit after we have evaluated expected sales and
leftover inventory. See Exhibit 12.5 for a summary of this procedure.
257
Exhibit 12.5
EXPECTED SALES, EXPECTED LEFTOVER INVENTORY, AND EXPECTED PROFIT
EVALUATION PROCEDURES
Step 1. Evaluate expected lost sales (see Exhibit 12.4). All of these performance mea-
Step 2. sures can be evaluated directly in terms of expected lost sales and several known
parameters: ϭ Expected demand; Q ϭ Order quantity; Price; Cost; and Salvage
value.
Use the following equations to evaluate the performance measure of interest.
Expected sales ϭ m Ϫ Expected lost sales
Expected leftover inventory ϭ Q Ϫ Expected sales
ϭ Q Ϫ m ϩ Expected lost sales
Expected profit ϭ [(Price Ϫ Cost) ϫ Expected sales]
Ϫ [(Cost Ϫ Salvage value) ϫ Expected leftover inventory]
With an order quantity of 3,500 units and a normal distribution demand forecast, the
expected profit for the Hammer 3/2 is
Expected profit ($80 2,858) ($20 × 642) $215, 800
In-Stock Probability and Stockout Probability
A common measure of customer service is the in-stock probability. The in-stock probability
is the probability the firm ends the season having satisfied all demand. (Equivalently, the
in-stock probability is the probability the firm has stock available for every customer.) That
occurs if demand is less than or equal to the order quantity,
In-stock probability ϭ F(Q)
The stockout probability is the probability the firm stocks out for some customer during the
selling season (i.e., a lost sale occurs). Because the firm stocks out if demand exceeds the
order quantity,
Stockout probability 1 F(Q)
(The firm either stocks out or it does not, so the stockout probability equals 1 minus the
probability demand is Q or lower.) We also can see that the stockout probability and the
in-stock probability are closely related:
Stockout probability 1 In-stock probability
See Exhibit 12.6 for a summary of the procedure to evaluate these probabilities. With an
order quantity of 3,500 Hammers, the z-statistic is z ϭ (3,500 − 3,192)/1,181 ϭ 0.26. From
the Standard Normal Distribution Function Table, we find ⌽(0.26) ϭ 0.6026, so the in-
stock probability is 60.26 percent. The stockout probability is 1 − 0.6026 ϭ 39.74 percent.
The in-stock probaility is not the only measure of customer service. Another popular
measure is the fill rate. The fill rate is the probability a customer is able to purchase a unit
(i.e., does not experience a stockout). Interestingly, this is not the same as the in-stock
probability, which is the probability that all demand is satisfied. For example, if Q ϭ 100
and demand turns out to be 101, then most customers were able to purchase a unit but the
firm did not satisfy all demand. See Appendix D for more information regarding how to
evaluate the fill rate.
258
Exhibit 12.6
IN-STOCK PROBABILITY AND STOCKOUT PROBABILITY EVALUATION
If the demand forecast is a normal distribution with mean and standard deviation , then
follow steps A through D:
A. Evaluate the z-statistic for the order quantity: z ϭ Q Ϫ m.
s
B. Use the z-statistic to look up in the Standard Normal Distribution Function Table the
probability the standard normal demand is z or lower, ⌽(z).
C. In-stock probability ϭ ⌽(z) and Stockout probability ϭ 1 − ⌽(z).
D. In Excel, In-stock probability ϭ Normsdist(z) and Stockout probability ϭ 1 − Normsdist(z).
If the demand forecast is a discrete distribution function table, then In-stock probability ϭ F(Q)
and Stockout probability ϭ 1 − F(Q), where F(Q) is the probability demand is Q or lower.
12.6 Choosing an Order Quantity to Meet a Service Objective
Maximizing expected profit is surely a reasonable objective for choosing an order quantity,
but it is not the only objective. As we saw in the previous section, the expected profit-
maximizing order quantity may generate an unacceptable in-stock probability from the
firm’s customer service perspective. This section explains how to determine an order
quantity that satisfies a customer service objective, in particular, a minimum in-stock
probability.
Suppose O’Neill wants to find the order quantity that generates a 99 percent in-stock
probability with the Hammer 3/2. The in-stock probability is F(Q). So we need to find an
order quantity such that there is a 99 percent probability that demand is that order quantity
or lower. Given that our demand forecast is normally distributed, we first find the z-statistic
that achieves our objective with the standard normal distribution. In the Standard Normal
Distribution Function Table, we see that ⌽(2.32) ϭ 0.9898 and ⌽(2.33) ϭ 0.9901. Again,
we choose the higher z-statistic, so our desired order quantity is now Q ϭ ϩ z ϫ ϭ
3,192 ϩ 2.33 ϫ 1,181 ϭ 5,944. You can use Excel to avoid looking up a probability in the
Standard Normal Distribution Function Table to find z:
z ϭ Normsinv(In-stock probability)
Notice that a substantially higher order quantity is needed to generate a 99 percent in-stock
probability than the one that maximizes expected profit (4,196). Exhibit 12.7 summarizes
the process for finding an order quantity to satisfy a target in-stock probability.
12.7 Managerial Lessons
Now that we have detailed the process of implementing the newsvendor model, it is worth-
while to step back and consider the managerial lessons it implies.
With respect to the forecasting process, there are three key lessons.
• For each product, it is insufficient to have just a forecast of expected demand. We
also need a forecast for how variable demand will be about the forecast. That uncertainty
in the forecast is captured by the standard deviation of demand.
259
Exhibit 12.7
A PROCEDURE TO DETERMINE AN ORDER QUANTITY THAT SATISFIES A TARGET
IN-STOCK PROBABILITY
If the demand forecast is a normal distribution with mean and standard deviation , then
follow steps A and B:
A. Find the z-statistic in the Standard Normal Distribution Function Table that satisfies the
in-stock probability, that is,
F(z) In-stock probability
If the in-stock probability falls between two z values in the table, choose the higher z. In
Excel, z can be found with the following formula:
z ϭ Normsinv(In-stock probability).
B. Convert the chosen z-statistic into the order quantity that satisfies our target in-stock
probability,
Q ϭ mϩz ϫs
If the demand forecast is a discrete distribution function table, then find the order quantity
in the table such that F(Q) ϭ In-stock probability. If the in-stock probability falls between
two entries in the table, choose the entry with the larger order quantity.
• It is important to track actual demand. Two common mistakes are made with respect
to this issue. First, do not forget that actual demand may be greater than actual sales due to
an inventory shortage. If it is not possible to track actual demand after a stockout occurs,
then you should attempt a reasonable estimate of actual demand. Second, actual demand
includes potential sales only at the regular price. If you sold 1,000 units in the previous
season, but 600 of them were at the discounted price at the end of the season, then actual
demand is closer to 400 than 1,000.
• You need to keep track of past forecasts and forecast errors in order to assess the
standard deviation of demand. Without past data on forecasts and forecast errors, it is
very difficult to choose reasonable standard deviations; it is hard enough to forecast the
mean of a distribution, but forecasting the standard deviation of a distribution is nearly
impossible with just a “gut feel.” Unfortunately, many firms fail to maintain the data
they need to implement the newsvendor model correctly. They might not record the data
because it is an inherently undesirable task to keep track of past errors: Who wants to
have a permanent record of the big forecasting goofs? Alternatively, firms may not real-
ize the importance of such data and therefore do not go through the effort to record and
maintain it.
There are also a number of important lessons from the order quantity choice process.
• The profit-maximizing order quantity generally does not equal expected demand. If
the underage cost is greater than the overage cost (i.e., it is more expensive to lose a sale
than it is to have leftover inventory), then the profit-maximizing order quantity is larger
than expected demand. (Because then the critical ratio is greater than 0.50.) On the other
hand, some products may have an overage cost that is larger than the underage cost. For
such products, it is actually best to order less than the expected demand.
260
Betting on Uncertain Demand: The Newsvendor Model 261
• The order quantity decision should be separated from the forecasting process. The
goal of the forecasting process is to develop the best forecast for a product’s demand
and therefore should proceed without regard to the order quantity decision. This can
be frustrating for some firms. Imagine the marketing department dedicates consider-
able effort to develop a forecast and then the operations department decides to produce
a quantity above the forecast. The marketing department may feel that their efforts
are being ignored or their expertise is being second-guessed. In addition, they may
be concerned that they would be responsible for ensuring that all of the production is
sold even though their forecast was more conservative. The separation between the
forecasting and the order quantity decision also implies that two products with the
same mean forecast may have different expected profit-maximizing order quantities,
either because they have different critical ratios or because they have different standard
deviations.
• Explicit costs should not be overemphasized relative to opportunity costs. Inventory
at the end of the season is the explicit cost of a demand–supply mismatch, while lost sales
are the opportunity cost. Overemphasizing the former relative to the latter will cause you
to order less than the profit-maximizing order quantity.
• It is important to recognize that choosing an order quantity to maximize expected
profit is only one possible objective. It is also a very reasonable objective, but there
can be situations in which a manager may wish to consider an alternative objective. For
example, maximizing expected profit is wise if you are not particularly concerned with
the variability of profit. If you are managing many different products so that the realized
profit from any one product cannot cause undue hardship on the firm, then maximizing
expected profit is a good objective to adopt. But if you are a startup firm with a single
product and limited capital, then you might not be able to absorb a significant profit loss.
In situations in which the variability of profit matters, it is prudent to order less than the
profit-maximizing order quantity. The expected profit objective also does not consider
customer service explicitly in its objective. With the expected profit-maximizing order
quantity for the Hammer 3/2, the in-stock probability is about 80 percent. Some man-
agers may feel this is an unacceptable level of customer service, fearing that unsatis-
fied customers will switch to a competitor. Figure 12.6 displays the trade-off between
FIGURE 12.6 Expected Profit 230
Thousands 225
The Trade-off 220
between Profit and 215
Service with the 210
Hammer 3/2 205
The circle indicates the 200
in-stock probability 195
and the expected profit 190
of the optimal order 185
quantity, 4,196 units. 180
65% 70% 75% 80% 85% 90% 95% 100%
In-stock Probabilty
262 Chapter 12
service and expected profit. As we can see, the expected profit curve is reasonably flat
around the maximum, which occurs with an in-stock probability that equals 80 percent.
Raising the in-stock probability to 90 percent may be considered worthwhile because it
reduces profits by slightly less than 1 percent. However, raising the in-stock dramati-
cally, say, to 99 percent, may cause expected profits to fall too much—in that case by
nearly 10 percent.
• Finally, while it is impossible to perfectly match supply and demand when supply
must be chosen before random demand, it is possible to make a smart choice that balances
the cost of ordering too much with the cost of ordering too little. In other words, uncer-
tainty should not invite ad hoc decision making.
12.8 The newsvendor model is a tool for making a decision when there is a “too much–too little”
Summary challenge: Bet too much and there is a cost (e.g., leftover inventory), but bet too little and
there is a different cost (e.g., the opportunity cost of lost sales). (See Table 12.4 for a sum-
mary of the key notation and equations.) To make this trade-off effectively, it is necessary
to have a complete forecast of demand. It is not enough to just have a single sales forecast;
we need to know the potential variation about that sales forecast. With a forecast model of
demand (e.g., normal distribution with mean 3,192 and standard deviation 1,181), we can
choose a quantity to maximize expected profit or to achieve a desired in-stock probability.
For any chosen quantity, we can evaluate several performance measures, such as expected
sales and expected profit.
TABLE 12.4 Q ϭ Order quantity Cu ϭ Underage cost Co ϭ Overage cost Critical ratio Cu
Summary of Key Co Cu
Notation and
Equations in ϭ Expected demand ϭ Standard deviation of demand
Chapter 12
F(Q): Distribution function ⌽(Q): Distribution function of the standard normal
Expected actual demand ϭ Expected A/F ratio ϫ Forecast
Standard deviation of actual demand ϭ Standard deviation of A/F ratios ϫ Forecast
Expected profit-maximizing order quantity: F (Q) Cu
Co Cu
z-statistic or normalized order quantity: z ϭ Q Ϫ m.
s
Qϭϩzϫ
L(z) ϭ Expected lost sales with the standard normal distribution
Expected lost sales ϭ ϫ L(z) Expected sales ϭ Ϫ Expected lost sales
Excel: Expected lost sales ϭ * (Normdist(z,0,1,0) Ϫ z * (1 Ϫ Normsdist(z)))
Expected leftover inventory ϭ Q Ϫ Expected sales
Expected profit ϭ [(Price Ϫ Cost) ϫ Expected sales]
Ϫ [(Cost Ϫ Salvage value) ϫ Expected leftover inventory]
In-stock probability ϭ F(Q) Stockout probability ϭ 1 Ϫ In-stock probability
Excel: z = Normsinv(Target in-stock probability)
Excel: In-stock probability ϭ Normsdist(z)
Betting on Uncertain Demand: The Newsvendor Model 263
12.9 The newsvendor model is one of the most extensively studied models in operations manage-
Further ment. It has been extended theoretically along numerous dimensions (e.g., multiple periods have
Reading been studied, the pricing decision has been included, the salvage values could depend on the quan-
tity salvaged, the decision maker’s tolerance for risk can be incorporated into the objective func-
12.10 tion, etc.)
Practice
Problems Several textbooks provide more technical treatments of the newsvendor model than this chapter.
See Nahmias (2005), Porteus (2002), or Silver, Pyke, and Peterson (1998).
For a review of the theoretical literature on the newsvendor model, with an emphasis on the pric-
ing decision in a newsvendor setting, see Petruzzi and Dada (1999).
Q12.1* (McClure Books) Dan McClure owns a thriving independent bookstore in artsy New
Q12.2* Hope, Pennsylvania. He must decide how many copies to order of a new book, Power
and Self-Destruction, an exposé on a famous politician’s lurid affairs. Interest in the
book will be intense at first and then fizzle quickly as attention turns to other celebri-
ties. The book’s retail price is $20 and the wholesale price is $12. The publisher will
buy back the retailer’s leftover copies at a full refund, but McClure Books incurs $4 in
shipping and handling costs for each book returned to the publisher. Dan believes his
demand forecast can be represented by a normal distribution with mean 200 and stan-
dard deviation 80.
a. Dan will consider this book to be a blockbuster for him if it sells more than 400 units.
What is the probability Power and Self-Destruction will be a blockbuster?
b. Dan considers a book a “dog” if it sells less than 50 percent of his mean forecast. What
is the probability this exposé is a “dog”?
c. What is the probability demand for this book will be within 20 percent of the mean
forecast?
d. What order quantity maximizes Dan’s expected profit?
e. Dan prides himself on good customer service. In fact, his motto is “McClure’s got what
you want to read.” How many books should Dan order if he wants to achieve a 95 percent
in-stock probability?
f. If Dan orders the quantity chosen in part e to achieve a 95 percent in-stock probability,
then what is the probability that “Dan won’t have what some customer wants to read”
(i.e., what is the probability some customer won’t be able to purchase a copy of the
book)?
g. Suppose Dan orders 300 copies of the book. What would Dan’s expected profit be in
this case?
(EcoTable Tea) EcoTable is a retailer of specialty organic and ecologically friendly
foods. In one of their Cambridge, Massachusetts, stores, they plan to offer a gift basket of
Tanzanian teas for the holiday season. They plan on placing one order and any leftover
inventory will be discounted at the end of the season. Expected demand for this store is
4.5 units and demand should be Poisson distributed. The gift basket sells for $55, the
purchase cost to EcoTable is $32, and leftover baskets will be sold for $20.
a. If they purchase only 3 baskets, what is the probability that some demand will not be
satisfied?
b. If they purchase 10 baskets, what is the probability that they will have to mark down at
least 3 baskets?
c. How many baskets should EcoTable purchase to maximize its expected profit?
d. Suppose they purchase 4 baskets. How many baskets can they expect to sell?
e. Suppose they purchase 6 baskets. How many baskets should they expect to have to
mark down at the end of the season?
(* indicates that the solution is in Appendix E)
264 Chapter 12
Q12.3* f. Suppose EcoTable wants to minimize its inventory while satisfying all demand with at
least a 90 percent probability. How many baskets should they order?
g. Suppose EcoTable orders 8 baskets. What is its expected profit?
(Pony Express Creations) Pony Express Creations Inc. (www.pony-ex.com) is a manu-
facturer of party hats, primarily for the Halloween season. (80 percent of their yearly
sales occur over a six-week period.) One of their popular products is the Elvis wig, com-
plete with sideburns and metallic glasses. The Elvis wig is produced in China, so Pony
Express must make a single order well in advance of the upcoming season. Ryan, the
owner of Pony Express, expects demand to be 25,000 and the following is his entire
demand forecast:
Q Prob(D ؍Q) F(Q) L(Q)
5,000 0.0181 0.0181 20,000
10,000 0.0733 0.0914 15,091
15,000 0.1467 0.2381 10,548
20,000 0.1954 0.4335
25,000 0.1954 0.6289 6,738
30,000 0.1563 0.7852 3,906
35,000 0.1042 0.8894 2,050
40,000 0.0595 0.9489
45,000 0.0298 0.9787 976
50,000 0.0132 0.9919 423
55,000 0.0053 0.9972 168
60,000 0.0019 0.9991
65,000 0.0006 0.9997 61
70,000 0.0002 0.9999 21
75,000 0.0001 1.0000
7
2
0
0
Prob(D ϭ Q) ϭ Probability demand D equals Q
F(Q) ϭ Probability demand is Q or lower
L(Q) ϭ Expected lost sales if Q units are ordered
Q12.4* The Elvis wig retails for $25, but Pony Express’s wholesale price is $12. Their produc-
tion cost is $6. Leftover inventory can be sold to discounters for $2.50.
a. Suppose Pony Express orders 40,000 Elvis wigs. What is the chance they have to liqui-
date 10,000 or more wigs with a discounter?
b. What order quantity maximizes Pony Express’s expected profit?
c. If Pony Express wants to have a 90 percent in-stock probability, then how many Elvis
wigs should be ordered?
d. If Pony Express orders 50,000 units, then how many wigs can they expect to have to
liquidate with discounters?
e. If Pony Express insists on a 100 percent in-stock probability for its customers, then
what is its expected profit?
(Flextrola) Flextrola, Inc., an electronics systems integrator, is planning to design a key
component for their next-generation product with Solectrics. Flextrola will integrate the
component with some software and then sell it to consumers. Given the short life cycles
of such products and the long lead times quoted by Solectrics, Flextrola only has one
opportunity to place an order with Solectrics prior to the beginning of its selling season.
Flextrola’s demand during the season is normally distributed with a mean of 1,000 and a
standard deviation of 600.
(* indicates that the solution is in Appendix E)
Betting on Uncertain Demand: The Newsvendor Model 265
Solectrics’ production cost for the component is $52 per unit and it plans to sell the
component for $72 per unit to Flextrola. Flextrola incurs essentially no cost associated
with the software integration and handling of each unit. Flextrola sells these units to
consumers for $121 each. Flextrola can sell unsold inventory at the end of the season in
a secondary electronics market for $50 each. The existing contract specifies that once
Flextrola places the order, no changes are allowed to it. Also, Solectrics does not accept
any returns of unsold inventory, so Flextrola must dispose of excess inventory in the
secondary market.
a. What is the probability that Flextrola’s demand will be within 25 percent of its forecast?
b. What is the probability that Flextrola’s demand will be more than 40 percent greater
than its forecast?
c. Under this contract, how many units should Flextrola order to maximize its expected profit?
For parts d through i, assume Flextrola orders 1,200 units.
d. What are Flextrola’s expected sales?
e. How many units of inventory can Flextrola expect to sell in the secondary electronics
market?
f. What is Flextrola’s expected gross margin percentage, which is (Revenue − Cost)/Revenue?
g. What is Flextrola’s expected profit?
h. What is Solectrics’ expected profit?
i. What is the probability that Flextrola has lost sales of 400 units or more?
j. A sharp manager at Flextrola noticed the demand forecast and became wary of assuming
that demand is normally distributed. She plotted a histogram of demands from previous
seasons for similar products and concluded that demand is better represented by the
log normal distribution. Figure 12.7 plots the density function for both the log normal
FIGURE 12.7 Density Function
Density Function Probability
.0012
.0011
.0010 Log Normal Distribution
.0009
.0008
.0007
.0006 Normal Distribution
.0005
.0004
.0003
.0002
.0001
0.0000 250 500 750 1,000 1,250 1,500 1,750 2,000 2,250 2,500
0 Demand
266 Chapter 12
Q12.5* and the normal distribution, each with mean of 1,000 and standard deviation of 600.
Figure 12.8 plots the distribution function for both the log normal and the normal.
Using the more accurate forecast (i.e., the log normal distribution), approximately
how many units should Flextrola order to maximize its expected profit?
(Fashionables) Fashionables is a franchisee of The Limited, the well-known retailer
of fashionable clothing. Prior to the winter season, The Limited offers Fashionables
the choice of five different colors of a particular sweater design. The sweaters are
knit overseas by hand, and because of the lead times involved, Fashionables will need
to order its assortment in advance of the selling season. As per the contracting terms
offered by The Limited, Fashionables also will not be able to cancel, modify, or reor-
der sweaters during the selling season. Demand for each color during the season is
normally distributed with a mean of 500 and a standard deviation of 200. Further, you
may assume that the demands for each sweater are independent of those for a different
color.
The Limited offers the sweaters to Fashionables at the wholesale price of $40 per
sweater and Fashionables plans to sell each sweater at the retail price of $70 per unit.
The Limited delivers orders placed by Fashionables in truckloads at a cost of $2,000 per
truckload. The transportation cost of $2,000 is borne by Fashionables. Assume unless
otherwise specified that all the sweaters ordered by Fashionables will fit into one truck-
load. Also assume that all other associated costs, such as unpacking and handling, are
negligible.
The Limited does not accept any returns of unsold inventory. However, Fashionables can sell
all of the unsold sweaters at the end of the season at the fire-sale price of $20 each.
FIGURE 12.8 Distribution Function
Distribution Function Probability
1.00
.90
Log Normal Distribution
.80
.70
.60
.50
.40
Normal Distribution
.30
.20
.10
.00 250 500 750 1,000 1,250 1,500 1,750 2,000 2,250 2,500
0 Demand
(* indicates that the solution is in Appendix E)
Betting on Uncertain Demand: The Newsvendor Model 267
a. How many units of each sweater type should Fashionables order to maximize its
expected profit?
b. If Fashionables wishes to ensure a 97.5 percent in-stock probability, what should its
order quantity be for each type of sweater?
For parts c and d, assume Fashionables orders 725 of each sweater.
c. What is Fashionables’ expected profit?
d. What is the stockout probability for each sweater?
e. Now suppose that The Limited announces that the unit of truckload capacity is 2,500
total units of sweaters. If Fashionables orders more than 2,500 units in total (actually,
from 2,501 to 5,000 units in total), it will have to pay for two truckloads. What now is
Fashionables’ optimal order quantity for each sweater?
Q12.6** (Teddy Bower Parkas) Teddy Bower is an outdoor clothing and accessories chain that
purchases a line of parkas at $10 each from its Asian supplier, TeddySports. Unfortunately,
at the time of order placement, demand is still uncertain. Teddy Bower forecasts that its
demand is normally distributed with mean of 2,100 and standard deviation of 1,200. Teddy
Bower sells these parkas at $22 each. Unsold parkas have little salvage value; Teddy Bower
simply gives them away to a charity.
a. What is the probability this parka turns out to be a “dog,” defined as a product that sells
less than half of the forecast?
b. How many parkas should Teddy Bower buy from TeddySports to maximize expected
profit?
c. If Teddy Bower wishes to ensure a 98.5 percent in-stock probability, how many parkas
should it order?
For parts d and e, assume Teddy Bower orders 3,000 parkas.
d. Evaluate Teddy Bower’s expected profit.
e. Evaluate Teddy Bower’s stockout probability
Q12.7 (Teddy Bower Boots) To ensure a full line of outdoor clothing and accessories, the
marketing department at Teddy Bower insists that they also sell waterproof hunting
boots. Unfortunately, neither Teddy Bower nor TeddySports has expertise in manufac-
turing those kinds of boots. Therefore, Teddy Bower contacted several Taiwanese sup-
pliers to request quotes. Due to competition, Teddy Bower knows that it cannot sell
these boots for more than $54. However, $40 per boot was the best quote from the sup-
pliers. In addition, Teddy Bower anticipates excess inventory will need to be sold off
at a 50 percent discount at the end of the season. Given the $54 price, Teddy Bower’s
demand forecast is for 400 boots, with a standard deviation of 300.
a. If Teddy Bower decides to include these boots in its assortment, how many boots
should it order from its supplier?
b. Suppose Teddy Bower orders 380 boots. What would its expected profit be?
Q12.8 c. John Briggs, a buyer in the procurement department, overheard at lunch a discussion of
the “boot problem.” He suggested that Teddy Bower ask for a quantity discount from
the supplier. After following up on his suggestion, the supplier responded that Teddy
Bower could get a 10 percent discount if they were willing to order at least 800 boots.
If the objective is to maximize expected profit, how many boots should it order given
this new offer?
(Land’s End) Geoff Gullo owns a small firm that manufactures “Gullo Sunglasses.” He
has the opportunity to sell a particular seasonal model to Land’s End. Geoff offers Land’s
End two purchasing options:
• Option 1. Geoff offers to set his price at $65 and agrees to credit Land’s End $53 for
each unit Land’s End returns to Geoff at the end of the season (because
those units did not sell). Since styles change each year, there is essentially
no value in the returned merchandise.
268 Chapter 12
Q12.9 • Option 2. Geoff offers a price of $55 for each unit, but returns are no longer accepted.
In this case, Land’s End throws out unsold units at the end of the season.
This season’s demand for this model will be normally distributed with mean of 200 and
standard deviation of 125. Land’s End will sell those sunglasses for $100 each. Geoff ’s
production cost is $25.
a. How much would Land’s End buy if they chose option 1?
b. How much would Land’s End buy if they chose option 2?
c. Which option will Land’s End choose?
d. Suppose Land’s End chooses option 1 and orders 275 units. What is Geoff Gullo’s
expected profit?
(CPG Bagels) CPG Bagels starts the day with a large production run of bagels. Through-
out the morning, additional bagels are produced as needed. The last bake is completed at
3 P.M. and the store closes at 8 P.M. It costs approximately $0.20 in materials and labor to
make a bagel. The price of a fresh bagel is $0.60. Bagels not sold by the end of the day
are sold the next day as “day old” bagels in bags of six, for $0.99 a bag. About two-thirds
of the day-old bagels are sold; the remainder are just thrown away. There are many bagel
flavors, but for simplicity, concentrate just on the plain bagels. The store manager predicts
that demand for plain bagels from 3 P.M. until closing is normally distributed with mean of
54 and standard deviation of 21.
a. How many bagels should the store have at 3 P.M. to maximize the store’s expected profit
(from sales between 3 P.M. until closing)? (Hint: Assume day-old bagels are sold for
$0.99/6 ϭ $0.165 each; i.e., don’t worry about the fact that day-old bagels are sold in
bags of six.)
b. Suppose that the store manager is concerned that stockouts might cause a loss of future
business. To explore this idea, the store manager feels that it is appropriate to assign
a stockout cost of $5 per bagel that is demanded but not filled. (Customers frequently
purchase more than one bagel at a time. This cost is per bagel demanded that is not
satisfied rather than per customer that does not receive a complete order.) Given the
additional stockout cost, how many bagels should the store have at 3 P.M. to maximize
the store’s expected profit?
c. Suppose the store manager has 101 bagels at 3 P.M. How many bagels should the store
manager expect to have at the end of the day?
Q12.10** (The Kiosk) Weekday lunch demand for spicy black bean burritos at the Kiosk, a local
snack bar, is approximately Poisson with a mean of 22. The Kiosk charges $4.00 for each
burrito, which are all made before the lunch crowd arrives. Virtually all burrito customers
also buy a soda that is sold for 60¢. The burritos cost the Kiosk $2.00, while sodas cost the
Kiosk 5¢. Kiosk management is very sensitive about the quality of food they serve. Thus,
they maintain a strict “No Old Burrito” policy, so any burrito left at the end of the day is
disposed of. The distribution function of a Poisson with mean 22 is as follows:
Q F(Q) Q F(Q) Q F(Q) Q F(Q)
1 0.0000 11 0.0076 21 0.4716 31 0.9735
2 0.0000 12 0.0151 22 0.5564 32 0.9831
3 0.0000 13 0.0278 23 0.6374 33 0.9895
4 0.0000 14 0.0477 24 0.7117 34 0.9936
5 0.0000 15 0.0769 25 0.7771 35 0.9962
6 0.0001 16 0.1170 26 0.8324 36 0.9978
7 0.0002 17 0.1690 27 0.8775 37 0.9988
8 0.0006 18 0.2325 28 0.9129 38 0.9993
9 0.0015 19 0.3060 29 0.9398 39 0.9996
10 0.0035 20 0.3869 30 0.9595 40 0.9998
Betting on Uncertain Demand: The Newsvendor Model 269
a. Suppose burrito customers buy their snack somewhere else if the Kiosk is out of stock.
How many burritos should the Kiosk make for the lunch crowd?
b. Suppose that any customer unable to purchase a burrito settles for a lunch of Pop-Tarts
and a soda. Pop-Tarts sell for 75¢ and cost the Kiosk 25¢. (As Pop-Tarts and soda are
easily stored, the Kiosk never runs out of these essentials.) Assuming that the Kiosk man-
agement is interested in maximizing profits, how many burritos should they prepare?
You can view a video of how problems marked with a ** are solved by going on www.
cachon-terwiesch.net and follow the links under ‘Solved Practice Problems’
13Chapter
Assemble-to-Order,
Make-to-Order, and
Quick Response with
Reactive Capacity1
A firm facing the newsvendor problem can manage, but not avoid, the possibility of a
demand–supply mismatch: order too much and inventory is left over at the end of the
season, but order too little and incur the opportunity cost of lost sales. The firm finds itself
in this situation because it commits to its entire supply before demand occurs. This mode
of operation is often called make-to-stock because all items enter finished goods inventory
(stock) before they are demanded. In other words, with make-to-stock, the identity of an
item’s eventual owner is not known when production of the item is initiated.
To reduce the demand–supply mismatches associated with make-to-stock, a firm could
attempt to delay at least some production until better demand information is learned. For
example, a firm could choose to begin producing an item only when it receives a firm order
from a customer. This mode of operation is often called make-to-order or assemble-to-
order. Dell Computer is probably the most well-known and most successful company to
have implemented the assemble-to-order model.
Make-to-stock and make-to-order are two extremes in the sense that with one all pro-
duction begins well before demand is received, whereas with the other production begins
only after demand is known. Between any two extremes there also must be an intermediate
option. Suppose the lead time to receive an order is short relative to the length of the sell-
ing season. A firm then orders some inventory before the selling season starts so that some
product is on hand at the beginning of the season. After observing early season sales, the
firm then submits a second order that is received well before the end of the season (due
to the short lead time). In this situation, the firm should make a conservative initial order
and use the second order to strategically respond to initial season sales: Slow-selling prod-
ucts are not replenished midseason, thereby reducing leftover inventory, while fast-selling
products are replenished, thereby reducing lost sales.
The capability to place multiple orders during a selling season is an integral part of Quick
Response. Quick Response is a set of practices designed to reduce the cost of mismatches
1 The data in this chapter have been modified to protect confidentiality.
270
Assemble-to-Order, Make-to-Order, and Quick Response with Reactive Capacity 271
between supply and demand. It began in the apparel industry as a response to just-in-time
practices in the automobile industry and has since migrated to the grocery industry under
the label Efficient Consumer Response.
The aspect of Quick Response discussed in this chapter is the use of reactive capacity,
that is, capacity that allows a firm to place one additional order during the season, which
retailers often refer to as a “second buy.” As in Chapter 12, we use O’Neill Inc. for our case
analysis. Furthermore, we assume throughout this chapter that the normal distribution with
mean 3,192 and standard deviation 1,181 is our demand forecast for the Hammer 3/2.
The first part of this chapter evaluates and minimizes the demand–supply mismatch cost
to a make-to-stock firm, that is, a firm that has only a single ordering opportunity, as in the
newsvendor model. Furthermore, we identify situations in which the cost of demand–supply
mismatches is large. Those are the situations in which there is the greatest potential to
benefit from Quick Response with reactive capacity or make-to-order production. The
second part of this chapter discusses make-to-order relative to make-to-stock. The third
part studies reactive capacity: How should we choose an initial order quantity when some
reactive capacity is available? And, as with the newsvendor model, how do we evaluate
several performance measures? The chapter concludes with a summary and managerial
implications.
13.1 Evaluating and Minimizing the Newsvendor’s
Demand–Supply Mismatch Cost
In this section, the costs associated in the newsvendor model with demand–supply mismatches
are identified, then two approaches are outlined for evaluating the expected demand–supply
mismatch cost, and finally we show how to minimize those costs. For ease of exposition,
we use the shorthand term mismatch cost to refer to the “expected demand–supply mis-
match cost.”
In the newsvendor model, the mismatch cost is divided into two components: the cost
of ordering too much and the cost of ordering too little. Ordering too much means there is
leftover inventory at the end of the season. Ordering too little means there are lost sales.
The cost for each unit of leftover inventory is the overage cost, which we label Co. The cost
for each lost sale is the underage cost, which we label Cu. (See Chapter 12 for the original
discussion of these costs.) Therefore, the mismatch cost in the newsvendor model is the
sum of the expected overage cost and the expected underage cost:
Mismatch cost (Co Expected leftover inventory) (13.1)
(Cu Expected lost sales)
Notice that the mismatch cost includes both a tangible cost (leftover inventory) and an
intangible opportunity cost (lost sales). The former has a direct impact on the profit and
loss statement, but the latter does not. Nevertheless, the opportunity cost of lost sales
should not be ignored.
Not only does equation (13.1) provide us with the definition of the mismatch cost,
it also provides us with our first method for evaluating the mismatch cost because we
already know how to evaluate the expected leftover inventory and the expected lost sales
(from Chapter 12). Let’s illustrate this method with O’Neill’s Hammer 3/2 wetsuit. The
Hammer has a selling price of $190 and a purchase cost from the TEC Group of $110.
Therefore, the underage cost is $190 Ϫ $110 ϭ $80 per lost sale. Leftover inventory is
sold at $90, so the overage cost is $110 Ϫ $90 ϭ $20 per wetsuit left at the end of the season.
The expected profit-maximizing order quantity is 4,196 units. Using the techniques
272 Chapter 13 Order quantity, Q ϭ 4,196 units
Expected demand, ϭ 3,192 units
TABLE 13.1 Standard deviation of demand, ϭ 1,181
Summary of Expected lost sales ϭ 130 units
Performance Expected sales ϭ 3,062 units
Measures for Expected leftover inventory ϭ 1,134 units
O’Neill’s Hammer Expected revenue ϭ $683,840
3/2 Wetsuit When Expected profit ϭ $222,280
the Expected Profit-
Maximizing Quantity Expected lost sales ϭ 1,181 ϫ L (0.85) ϭ 1,181 ϫ 0.11 ϭ 130
Is Ordered and the Expected sales ϭ 3,192 Ϫ 130 ϭ 3,062
Demand Forecast Is Expected leftover inventory ϭ 4,196 Ϫ 3,062 ϭ 1,134
Normally Distributed Expected revenue ϭ Price ϫ Expected sales ϩ Salvage value ϫ Expected leftover inventory
with Mean 3,192 and
Standard Deviation ϭ $190 ϫ 3,062 ϩ $90 ϫ 1,134 ϭ $683,840
1,181 Expected profit ϭ ($190 Ϫ $110) ϫ 3,062 Ϫ ($110 Ϫ $90) ϫ 1,134 ϭ $222,280
described in Chapter 12, for that order quantity we can evaluate several performance
measures, summarized in Table 13.1. Therefore, the mismatch cost for the Hammer 3/2,
despite ordering the expected profit-maximizing quantity, is
($20 1, 134) ($80 130) $33, 080
Now let’s consider a second approach for evaluating the mismatch cost. Imagine O’Neill
had the opportunity to purchase a magic crystal ball. Even before O’Neill needs to sub-
mit its order to TEC, this crystal ball reveals to O’Neill the exact demand for the entire
season. O’Neill would obviously order from TEC the demand quantity observed with this
crystal ball. As a result, O’Neill would be in the pleasant situation of avoiding all mis-
match costs (there would be no excess inventory and no lost sales) while still providing
immediate product availability to its customers. In fact, the only function of the crystal ball
is to eliminate all mismatch costs: for example, the crystal ball does not change demand,
increase the selling price, or decrease the production cost. Thus, the difference in O’Neill’s
expected profit with the crystal ball and without it must equal the mismatch cost: The crys-
tal ball increases profit by eliminating mismatch costs, so the profit increase must equal
the mismatch cost. Therefore, we can evaluate the mismatch cost by first evaluating the
newsvendor’s expected profit, then evaluating the expected profit with the crystal ball, and
finally taking the difference between those two figures.
We already know how to evaluate the newsvendor’s expected profit (again, see Chapter 12).
So let’s illustrate how to evaluate the expected profit with the crystal ball. If O’Neill gets to
observe demand before deciding how much to order from TEC, then there will not be any left-
over inventory at the end of the season. Even better, O’Neill will not stock out, so every unit
of demand turns into an actual sale. Hence, O’Neill’s expected sales with the crystal ball equal
expected demand, which is . We already know that O’Neill’s profit per sale is the gross
margin, the retail price minus the production cost, Price Ϫ Cost. Therefore O’Neill’s expected
profit with this crystal ball is expected demand times the profit per unit of demand, which is
(Price Ϫ Cost) ϫ . In fact, O’Neill can never earn a higher expected profit than it does with
the crystal ball: There is nothing better than having no leftover inventory and earning the full
margin on every unit of potential demand. Hence, let’s call that profit the maximum profit:
Maximum profit (Price Cost)
O’Neill’s maximum profit with the Hammer 3/2 is $80 ϫ 3,192 ϭ $255,360. We already
know that the newsvendor expected profit is $222,280. So the difference between the
Assemble-to-Order, Make-to-Order, and Quick Response with Reactive Capacity 273
maximum profit (i.e., crystal ball profit) and the newsvendor expected profit is O’Neill’s
mismatch costs. That figure is $255,360 Ϫ $222,280 ϭ $33,080, which matches our cal-
culation with our first method (as it should). To summarize, our second method for evalu-
ating the mismatch cost uses the following equation:
Mismatch cost Maximum profit Expected profit
Incidentally, you can also think of the mismatch cost as the most O’Neill should be willing
to pay to purchase the crystal ball; that is, it is the value of perfect demand information.
The second method for calculating the mismatch cost emphasizes that there exists an
easily evaluated maximum profit. We might not be able to evaluate expected profit pre-
cisely if there is some reactive capacity available to the firm. Nevertheless, we do know
that no matter what type of reactive capacity the firm has, that reactive capacity cannot be
as good as the crystal ball we just described. Therefore, the expected profit with any form
of reactive capacity must be more than the newsvendor’s expected profit but less than the
maximum profit.
You now may be wondering about how to minimize the mismatch cost and whether
that is any different than maximizing the newsvendor’s expected profit. The short
answer is that these are effectively the same objective, that is, the quantity that maxi-
mizes profit also minimizes mismatch costs. One way to see this is to look at the equa-
tion above: If expected profit is maximized and the maximum profit does not depend on
the order quantity, then the difference between them, which is the mismatch cost, must
be minimized.
Now that we know how to evaluate and minimize the mismatch cost, we need to get a
sense of its significance. In other words, is $33,080 a big problem or a little problem? To
answer that question, we need to compare it with something else. The maximum profit is one
reference point: the demand–supply mismatch cost as a percentage of the maximum profit
is $33,080/$255,360 ϭ 13 percent. You may prefer expected sales as a point of comparison:
the demand–supply mismatch cost per unit of expected sales is $33,080/3,062 ϭ $10.8.
Alternatively, we can make the comparison with expected revenue, $683,840, or expected
profit, $222,280: the demand–supply mismatch cost is approximately 4.8 percent of total
revenue ($33,080/$683,840) and 14.9 percent of expected profit ($33,080/$222,280). Com-
panies in the sports apparel industry generally have net profit in the range of 2 to 5 percent
of revenue. Therefore, eliminating the mismatch cost from the Hammer 3/2 could poten-
tially double O’Neill’s net profit! That is an intriguing possibility.
13.2 When Is the Mismatch Cost High?
No matter which comparison you prefer, the mismatch cost for O’Neill is significant,
even if the expected profit-maximizing quantity is ordered. But it is even better to know
what causes a large demand–supply mismatch. To answer that question, let’s first choose
our point of comparison for the mismatch cost. Of the ones discussed at the end of the
previous section, only the maximum profit does not depend on the order quantity chosen:
unit sales, revenue, and profit all clearly depend on Q. In addition, the maximum profit
is representative of the potential for the product: we cannot do better than earn the maxi-
mum profit. Therefore, let’s evaluate the mismatch cost as a percentage of the maximum
profit.
We next need to make an assumption about how much is ordered before the selling
season, that is, clearly the mismatch cost depends on the order quantity Q. Let’s adopt
274 Chapter 13
the natural assumption that the expected profit-maximizing quantity is ordered, which,
as we discussed in the previous section, also happens to minimize the newsvendor’s
mismatch cost.
If we take the equations for expected lost sales and expected leftover inventory from
Chapter 12, plug them into our first mismatch cost equation (13.1), and then do several
algebraic manipulations, we arrive at the following observations:
• The expected demand–supply mismatch cost becomes larger as demand variability
increases, where demand variability is measured with the coefficient of variation, /.
• The expected demand–supply mismatch cost becomes larger as the critical ratio,
Cu /(Co ϩ Cu), becomes smaller.
(If you want to see the actual equations and how they are derived, see Appendix D.)
It is intuitive that the mismatch cost should increase as demand variability increases—
it is simply harder to get demand to match supply when demand is less predicable. The
key insight is how to measure demand variability. The coefficient of variation is the cor-
rect measure. You may recall in Chapter 8 we discussed the coefficient of variation with
respect to the variability of the processing time (CVp) or the interarrival time to a queue
(CVa). This coefficient of variation, /, is conceptually identical to those coefficients
of variation: it is the ratio of the standard deviation of a random variable (in this case
demand) to its mean.
It is worthwhile to illustrate why the coefficient of variation is the appropriate measure
of variability in this setting. Suppose you are informed that the standard deviation of
demand for an item is 800. Does that tell you enough information to assess the variability
of demand? For example, does it allow you to evaluate the probability actual demand will
be less than 75 percent of your forecast? In fact, it does not. Consider two cases, in the first
the forecast is for 1,000 units and in the second the forecast is for 10,000 units. Demand is
less than 75 percent of the 1,000-unit forecast if demand is less than 750 units. What is the
probability that occurs? First, normalize the value 750:
ZQ 750 1, 000 0.31
800
Now use the Standard Normal Distribution Function Table to find the probability demand
is less than 750: ⌽(Ϫ0.31) ϭ 0.3783. With the forecast of 10,000, the comparable event
has demand that is less than 7,500 units. Repeating the same process yields z ϭ (7,500
Ϫ 10,000)/800 ϭ Ϫ3.1 and ⌽(Ϫ3.1) ϭ 0.0009. Therefore, with a standard deviation of
800, there is about a 38 percent chance demand is less than 75 percent of the first forecast
but much less than a 1 percent chance demand is less than 75 percent of the second forecast.
In other words, the standard deviation alone does not capture how much variability there
is in demand. Notice that the coefficient of variation with the first product is 0.8 (800/1,000),
whereas it is much lower with the second product, 0.08 (800/10,000).
For the Hammer 3/2, the coefficient of variation is 1,181/3,192 ϭ 0.37. While there is no
generally accepted standard for what is a “low,” “medium,” or “high” coefficient of variation,
we offer the following guideline: Demand variability is rather low if the coefficient of variation
is less than 0.25, medium if it is in the range 0.25 to 0.75, and high with anything above 0.75. A
coefficient of variation above 1.5 is extremely high, and anything above 3 would imply that the
demand forecast is essentially meaningless.
Table 13.2 provides data to allow you to judge for yourself what is a “low,” “medium,”
and “high” coefficient of variation.
Assemble-to-Order, Make-to-Order, and Quick Response with Reactive Capacity 275
TABLE 13.2 Coefficient Probability Demand Is Probability Demand Is
Forecast Accuracy of Variation Less Than 75% of the Forecast within 25% of the Forecast
Relative to the
Coefficient of 0.10 0.6% 98.8%
Variation When 0.25 15.9 68.3
Demand Is Normally 0.50 30.9 38.3
Distributed 0.75 36.9 26.1
1.00 40.1 19.7
1.50 43.4 13.2
2.00 45.0
9.9
3.00 46.7
6.6
Recall from Chapters 8 and 9 that the coefficient of variation with an exponential
distribution is always one. Therefore, if two processes have exponential distributions, they
always have the same amount of variability. The same is not true with the normal distribution
because with the normal distribution the standard deviation is adjustable relative to the mean.
Our second observation above relates mismatch costs to the critical ratio. In particu-
lar, products with low critical ratios and high demand variability have high mismatch
costs and products with high critical ratios and low demand variability have low mis-
match costs. Table 13.3 displays data on the mismatch cost for various coefficients of
variation and critical ratios.
As we have already mentioned, it is intuitive that the mismatch cost should increase
as demand variability increases. The intuition with respect to the critical ratio takes some
more thought. A very high critical ratio means there is a large profit margin relative to
the loss on each unit of excess inventory. Greeting cards are good examples of products
that might have very large critical ratios: the gross margin on each greeting card is large
while the production cost is low. With a very large critical ratio, the optimal order quan-
tity is quite large, so there are very few lost sales. There is also a substantial amount of
leftover inventory, but the cost of each unit left over in inventory is not large at all, so
the total cost of leftover inventory is relatively small. Therefore, the total mismatch cost
is small. Now consider a product with a low critical ratio, that is, the per-unit cost of
excess inventory is much higher than the cost of each lost sale. Perishable items often fall
into this category as well as items that face obsolescence. Given that excess inventory is
expensive, the optimal order quantity is quite low, possibly lower than expected demand.
As a result, excess inventory is not a problem, but lost sales are a big problem, resulting
in a high mismatch cost.
TABLE 13.3 Critical Ratio
The Mismatch Cost
(as a Percentage of Coefficient of Variation 0.4 0.5 0.6 0.7 0.8 0.9
the Maximum Profit)
When Demand Is 0.10 10% 8% 6% 5% 3% 2%
Normally Distributed 0.25 24% 20% 16% 12% 9% 5%
and the Newsvendor 0.40 39% 32% 26% 20% 14% 8%
Expected Profit- 0.55 53% 44% 35% 27% 19% 11%
Maximizing Quantity 0.70 68% 56% 45% 35% 24% 14%
Is Ordered 0.85 82% 68% 55% 42% 30% 17%
1.00 97% 80% 64% 50% 35% 19%
276 Chapter 13
13.3 Reducing Mismatch Costs with Make-to-Order
When supply is chosen before demand is observed (make-to-stock), there invariably is
either too much or too little supply. A purely hypothetical solution to the problem is to find
a crystal ball that reveals demand before it occurs. A more realistic solution is to initiate
production of each unit only after demand is observed for that unit, which is often called
make-to-order or assemble-to-order. This section discusses the pros and cons of make-to-
order with respect to its ability to reduce mismatch costs.
In theory, make-to-order can eliminate the entire mismatch cost associated with make-
to-stock (i.e., newsvendor). With make-to-order, there is no leftover inventory because
production only begins after a firm order is received from a customer. Thus, make-to-order
saves on expensive markdown and disposal expenses. Furthermore, there are no lost sales
with make-to-order because each customer order is eventually produced. Therefore, prod-
ucts with a high mismatch cost (low critical ratios, high demand variability) would benefit
considerably from a switch to make-to-order from make-to-stock.
But there are several reasons to be wary of make-to-order. For one, even with make-to-
order, there generally is a need to carry component inventory. Although components may
be less risky than finished goods, there still is a chance of having too many or too few of
them. Next, make-to-order is never able to satisfy customer demands immediately; that
is, customers must wait to have their order filled. If the wait is short, then demand with
make-to-order can be nearly as high as with make-to-stock. But there is also some thresh-
old beyond which customers do not wait. That threshold level depends on the product:
customers are generally less willing to wait for diapers than they are for custom sofas.
It is helpful to think of queuing theory (Chapters 8 and 9) to understand what deter-
mines the waiting time with make-to-order. No matter the number of servers, a key charac-
teristic of a queuing system is that customer service begins only after a customer arrives to
the system, just as production does not begin with make-to-order until a customer commits
to an order. Another important feature of a queuing system is that customers must wait to
be processed if all servers are busy, just as a customer must wait with make-to-order if the
production process is working on the backlog of orders from previous customers.
To provide a reference point for this discussion, suppose O’Neill establishes a make-to-
order assembly line for wetsuits. O’Neill could keep in inventory the necessary raw materials
to fabricate wetsuits in a wide array of colors, styles, and quality levels. Wetsuits would then be
produced as orders are received from customers. The assembly line has a maximum production
rate, which would correspond to the service rate in a queue. Given that demand is random, the
interarrival times between customer orders also would be random, just as in a queuing system.
A key insight from queuing is that a customer’s expected waiting time depends nonlin-
early (a curve, not a straight line) on the system’s utilization (the ratio of the flow rate to
capacity): As the utilization approaches 100 percent, the waiting time approaches infinity.
(See Figure 8.21.) As a result, if O’Neill wishes to have a reasonably short waiting time for
customers, then O’Neill must be willing to operate with less than 100 percent utilization,
maybe even considerably less than 100 percent. Less than 100 percent utilization implies
idle capacity; for example, if the utilization is 90 percent, then 10 percent of the time the
assembly line is idle. Therefore, even with make-to-order production, O’Neill experiences
demand–supply mismatch costs. Those costs are divided into two types: idle capacity and
lost sales from customers who are unwilling to wait to receive their product. When com-
paring make-to-stock with make-to-order, you could say that make-to-order replaces the
cost of leftover inventory with the cost of idle capacity. Whether or not make-to-order is
preferable depends on the relative importance of those two costs.
Assemble-to-Order, Make-to-Order, and Quick Response with Reactive Capacity 277
While a customer’s expected waiting time may be significant, customers are ultimately con-
cerned with their total waiting time, which includes the processing time. With make-to-order,
the processing time has two components: the time in production and the time from production
to actual delivery. Hence, successful implementation of make-to-order generally requires fast
and easy assembly of the final product. Next, keeping the delivery time to an acceptable level
either requires paying for fast shipping (e.g., air shipments) or moving production close to cus-
tomers (to reduce the distance the product needs to travel). Fast shipping increases the cost of
every unit produced, and local production (e.g., North America instead of Asia) may increase
labor costs. See Chapter 19 for more discussion.
Although make-to-order is not ideal for all products, Dell discovered that make-to-
order is particularly well suited for personal computers for several reasons: Inventory is
very expensive to hold because of obsolescence and falling component prices; labor is a
small portion of the cost of a PC, in part because the modular design of PCs allows for fast
and easy assembly; customers are primarily concerned with price and customization and
less concerned with how long they must wait for delivery (i.e., they are patient) and unique
design features (i.e., it is hard to differentiate one PC from another with respect to design);
there is a large pool of educated customers who are willing to purchase without physically
seeing the product (i.e., the phone/Internet channels work); and the cost to transport a PC
is reasonable (relative to its total value). The same logic suggests that make-to-order is
more challenging in the automobile industry. For example, assembling a vehicle is chal-
lenging, customization is less important to consumers, consumers do not like to wait to
receive their new vehicle (at least in the United States), and moving vehicles around is
costly (relative to their value). Indeed, Toyota once announced that it planned to produce a
custom-ordered vehicle in only five days (Simison 1999). However, the company quietly
backed away from the project.
As already mentioned, make-to-order is not ideal for all products. Koss Corp., a head-
phone maker, is an example of a company that discovered that make-to-order is not
always a magic bullet (Ramstad 1999). The company experimented with make-to-order
and discovered it was unable to provide timely deliveries to its customers (retailers) dur-
ing its peak season. In other words, demand was variable, but Koss’s capacity was not
sufficiently flexible. Because it began to lose business due to its slow response time,
Koss switched back to make-to-stock so that it would build up inventory before its peak
demand period. For Koss, holding inventory was cheaper than losing sales to impatient
customers. To summarize, make-to-order eliminates some of the demand–supply mis-
matches associated with make-to-stock, but make-to-order has its own demand–supply
mismatch issues. For example, make-to-order eliminates leftover inventory but it still
carries component inventory. More importantly, to ensure acceptable customer waiting
times, make-to-order requires some idle capacity, thereby potentially increasing labor and
delivery costs.
13.4 Quick Response with Reactive Capacity
O’Neill may very well conclude that make-to-order production is not viable either in Asia
(due to added shipping expenses) or in North America (due to added labor costs). If pure
make-to-order is out of the question, then O’Neill should consider some intermediate solu-
tion between make-to-stock (the newsvendor) and make-to-order (a queue). With the news-
vendor model, O’Neill commits to its entire supply before any demand occurs; whereas with
make-to-order, O’Neill commits to supply only after all demand occurs. The intermediate
solution is to commit to some supply before demand but then maintain the option to produce
278 Chapter 13
additional supply after some demand is observed. The capacity associated with that later
supply is called reactive capacity because it allows O’Neill to react to the demand informa-
tion it learns before committing to the second order. The ability to make multiple replenish-
ments (even if just one replenishment) is a central goal in Quick Response.
Suppose O’Neill approaches TEC with the request that TEC reduce its lead time.
O’Neill’s motivation behind this request is to try to create the opportunity for a replen-
ishment during the selling season. Recall that the Spring season spans six months, start-
ing in February and ending in July. (See Figure 12.2.) It has been O’Neill’s experience
that a hot product in the first two months of the season (i.e., a product selling above
forecast) almost always turns out to be a hot product in the rest of the season. As a
result, O’Neill could surely benefit from the opportunity to replenish the hot products
midseason. For example, suppose TEC offered a one-month lead time for a midseason
order. Then O’Neill could submit to TEC a second order at the end of the second month
(March) and receive that replenishment before the end of the third month, thereby allow-
ing that inventory to serve demand in the second half of the season. Figure 13.1 provides
a time line in this new situation.
While it is clear that O’Neill could benefit from the second order, offering a second order
with a one-month lead time can be costly to TEC. For example, TEC might need to reserve
some capacity to respond to O’Neill’s order. If O’Neill’s second order is not as large as TEC
anticipated, then some of that reserved capacity might be lost. Or O’Neill’s order might be
larger than anticipated, forcing TEC to scramble for extra capacity, at TEC’s expense. In
addition, the one-month lead time may force the use of faster shipping, which again could
increase costs. The issue is whether the cost increases associated with the second order jus-
tify the mismatch cost savings for O’Neill. To address this issue, let’s suppose that TEC
agrees to satisfy O’Neill’s second order but insists on a 20 percent premium for those units
to cover TEC’s anticipated additional expenses. Given this new opportunity, how should
O’Neill adjust its initial order quantity and how much are mismatch costs reduced?
Choosing order quantities with two ordering opportunities is significantly more complex
than choosing a single order quantity (i.e., the newsvendor problem). For instance, in addi-
tion to our forecast for the entire season’s demand, now we need to worry about developing
a forecast for demand in the second half of the season given what we observe in the first
two months of the season. Furthermore, we do not know what will be our initial sales when
we submit our first order, so that order must anticipate all possible outcomes for initial sales
and then the appropriate response in the second order for all of those outcomes. In addition,
we may stock out within the first half of the season if our first order is not large enough.
FIGURE 13.1 Generate Forecast Receive First Order Spring Selling Season
of Demand and from TEC at the (Feb. – July)
Time Line of Events Submit First End of January
for O’Neill’s Hammer Order to TEC
3/2 Wetsuit with
Unlimited, but Oct. Nov. Dec. Jan. Feb. Mar. Apr. May June July Aug.
Expensive, Reactive
Capacity
Observe Feb. and Receive Leftover
Mar. Sales and Second Units are
Submit Second Order from Discounted
Order to TEC TEC at the
End of April
Assemble-to-Order, Make-to-Order, and Quick Response with Reactive Capacity 279
Finally, even after observing initial sales, some uncertainty remains regarding demand in
the second half of the season.
Even though we now face a complex problem, we should not let the complexity overwhelm
us. A good strategy when faced with a complex problem is to make it less complex, that
is, make some simplifying assumptions that allow for analytical tractability while retain-
ing the key qualitative features of the complex problem. With that strategy in mind, let’s
assume (1) we do not run out of inventory before the second order arrives and (2) after we
observe initial sales we are able to perfectly predict sales in the remaining portion of the
season. Assumption 1 is not bad as long as the first order is reasonably large, that is, large
enough to cover demand in the first half of the season with a high probability. Assump-
tion 2 is not bad if initial sales are a very good predictor of subsequent sales, which has
been empirically observed in many industries.
Our simplifying assumptions are enough to allow us to evaluate the optimal initial order
quantity and then to evaluate expected profit. Let’s again consider O’Neill’s initial order
for the Hammer 3/2. It turns out that O’Neill still faces the “too much–too little” problem
associated with the newsvendor problem even though O’Neill has the opportunity to make
a second order. To explain, note that if the initial order quantity is too large, then there will
be leftover inventory at the end of the season. The second order does not help at all with
the risk of excess inventory, so the “too much” problem remains.
We also still face the “too little” issue with our initial order, but it takes a different form
than in our original newsvendor problem. Recall, with the original newsvendor problem,
ordering too little leads to lost sales. But the second order prevents lost sales: After we
observe initial sales, we are able to predict total demand for the remainder of the season.
If that total demand exceeds our initial order, we merely choose a second order quantity
to ensure that all demand is satisfied. This works because of our simplifying assumptions:
Lost sales do not occur before the second order arrives, there is no quantity limit on the
second order, and initial sales allow us to predict total demand for the season.
Although the second order opportunity eliminates lost sales, it does not mean we should
not bother with an initial order. Remember that units ordered during the season are more
expensive than units ordered before the season. Therefore, the penalty for ordering too
little in the first order is that we may be required to purchase additional units in the second
order at a higher cost.
Given that the initial order still faces the “too little–too much” problem, we can actually
use the newsvendor model to find the order quantity that maximizes expected profit. The
overage cost, Co, per unit of excess inventory is the same as in the original model; that is,
the overage cost is the loss on each unit of excess inventory. Recall that for the Hammer
3/2 Cost ϭ 110 and Salvage value ϭ 90. So Co ϭ 20.
The underage cost, Cu, per unit of demand that exceeds our initial order quantity is
the additional premium we must pay to TEC for units in the second order. That premium
is 20 percent, which is 20% ϫ 110 ϭ 22. In other words, if demand exceeds our initial
order quantity, then the penalty for ordering too little is the extra amount we must pay
TEC for each of those units (i.e., we could have avoided that premium by increasing the
initial order). Even though we must pay this premium to TEC, we are still better off hav-
ing the second ordering opportunity: Paying TEC an extra $22 for each unit of demand
that exceeds our initial order quantity is better than losing the $80 margin on each of those
units if we did not have the second order. So Cu ϭ 22.
We are now ready to calculate our optimal initial order quantity. (See Exhibit 12.3 for
an outline of this process.) First, evaluate the critical ratio:
Cu 22 0.5238
Co Cu 20 22
280 Chapter 13
Next find the z value in the Standard Normal Distribution Function Table that corresponds
to the critical ratio 0.5238: ⌽(0.05) ϭ 0.5199 and ⌽(0.06) ϭ 0.5239, so let’s choose the
higher z value, z ϭ 0.06. Now convert the z value into an order quantity for the actual
demand distribution with ϭ 3,192 and ϭ 1,181:
Q z 3,192 0.06 1,181 3, 263
Therefore, O’Neill should order 3,263 Hammer 3/2s in the first order to maximize expected
profit when a second order is possible. Notice that O’Neill should still order a considerable
amount in its initial order so as to avoid paying TEC the 20 percent premium on too many
units. However, O’Neill’s initial order of 3,263 units is considerably less than its optimal
order of 4,196 units when the second order is not possible.
Even though O’Neill must pay a premium with the second order, O’Neill’s expected
profit should increase by this opportunity. (The second order does not prevent O’Neill
from ordering 4,196 units in the initial order, so O’Neill cannot be worse off.) Let’s
evaluate what that expected profit is for any initial order quantity Q. Our maximum
profit has not changed. The best we can do is earn the maximum gross margin on every
unit of demand,
Maximum profit (Price Cost) (190 110) 3,192 255, 360
The expected profit is the maximum profit minus the mismatch costs:
Expected profit Maximum profit (Co Expected leftover inventory)
(Cu Expected second order quantity)
The first mismatch cost is the cost of leftover inventory and the second is the additional
premium that O’Neill must pay TEC for all of the units ordered in the second order. We
already know how to evaluate expected leftover inventory for any initial order quantity.
(See Exhibit 12.5 for a summary.) We now need to figure out the expected second order
quantity.
If we order Q units in the first order, then we make a second order only if demand
exceeds Q. In fact, our second order equals the difference between demand and Q, which
would have been our lost sales if we did not have a second order. This is also known as the
loss function. Therefore,
Expected second order quantity ϭ Newsvendor,s expected lost sales
We already know how to evaluate the newsvendor’s expected lost sales. (See Exhibit 12.4
for a summary.) First look up L(z) in the Standard Normal Loss Function Table for the z value
that corresponds to our order quantity, z ϭ 0.06. We find in that table L(0.06) ϭ 0.3697.
Next, finish the calculation:
Expected lost sales L(z) 1,181 0.3697 437
Recall that
Expected sales Expected lost sales 3,192 437 2, 755
where expected sales is the quantity the newsvendor would sell with an order quantity of
3,263. We want to evaluate expected sales for the newsvendor so that we can evaluate the
last piece we need:
Expected leftover inventory Q Expected sales 3, 263 2, 755 508
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